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2024-03-13 01:21:55

COS - Contemporary Style, Designed To Last - Shop Online - WW

SPRING SUMMER 2024 COLLECTION | WOMEN MEN

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SPRING SUMMER 2024

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COS | Luxury Fashion | ZALORA Philippines

| Luxury Fashion | ZALORA Philippines Login / RegisterWomenMenLuxurySportsKidsBeautyHome & LifestyleTrending Searches:

Women's Shirt Dresses

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COS Quilted Bag

COS: Minimalist Staples for the Modern Man & Woman

Where fashion trends come and go, COS remains steadfast in its goal to create timeless pieces that go beyond seasons. Focusing on creating minimalist staples that hinge on form, function, and everlasting fashion sense, this clothing brand for men and women is beloved for its superior quality, craftsmanship, and elegance that knows no season. With high-quality pieces that are versatile and easy to use in various outfits, COS provides a wide assortment of apparel such as neutral to bold tops, bottoms, bags, accessories, and shoes that let you bring out your true fashion persona. Browse through COS for men and women for simplistic yet standout pieces we know you’ll be reaching for time and time again.

Timeless Fashion That Lasts

Creating clothing that is effortlessly enduring through this season and beyond is not an easy feat. Thankfully, COS is here to bring everything you need for fashion that doesn’t go out of style. Peruse their selection of men's and women’s clothing for beautiful wardrobe essentials with modern silhouettes, sleek lines, and clean designs that are easy to wear, pair, and keep for years. Find COS dresses, blouses, jumpers, knitwear, trousers, cardigans, and more in various patterns and prints made from quality cotton and sustainable materials. Looking to get new workwear? Try a relaxed or fitted dress or culottes for the ladies or tapered trousers for the gentlemen. Top it off with a smart jacket, coats, and other outerwear for the ultimate corporate attire. Dress down for the weekend with casual denim jeans, joggers, hoodies, oversized t-shirts, shorts, and skirts. Complete your ensemble with a COS bag! From shopper bags to quilted bags to crossbody bags to sleek leather shoulder bags, you can look even more sophisticated with a bag from COS. Whatever the occasion, COS can provide the clothing essentials you’ll need for an elegant, fashion-forward wardrobe that lasts.

Contemporary With A Twist

Offering a refined yet contemporary aesthetic, COS creates minimalist womenswear staples and menswear essentials for modern creatives. Founded in 2007 as a diffusion brand of the H&M group, COS takes inspiration from Scandinavian simplicity, architecture, and art direction. The result is elegant, cleanly tailored pieces that transcend fast fashion trends. Step into any of COS's boutiques in Manila or shop online to discover their collection of versatile basics for any Pinay stylist on the move. Structured belted shirtdresses, draped blouses, and wide-leg culottes convey polished style for the creative workplace. Architectural shapes also carry through to knits, shirting, and outerwear rendered in textural natural fabrics. For the gents, COS delivers sharp essentials like slim wool trousers, fitted merino crews, and boxy canvas jackets suited for gallery openings or weekend pop-up markets. Luxe touches like silk scarves and sleek leather accessories complete the COS uniform with quiet sophistication.Each season, expect new minimalist renditions on classic garments and accessories featuring considered details and ethically conscious fabrics.Build an everyday transitional wardrobe anchored in COS's singular approach to wearable, design-driven style created responsibly with longevity in mind.

What Does COS Stand For?

COS, which stands for "Collection of Style", is a minimalist and sustainable Swedish fashion brand established in 2007. Now, COS is owned by the H&M group. Other brands under H&M include Monki, & Other Stories, Weekday, and ARKET.

Shop The Best Deals On COS Clothing Online At ZALORA Philippines

Get all of your modern wardrobe essentials for men and women online here with COS. Dress elegantly from head to toe with a wide assortment of tops, bottoms, outerwear, shoes, bags, and accessories from COS. As the leading online fashion and lifestyle destination in Asia, ZALORA has endless style possibilities thanks to an ever-expanding range of clothes, shoes, bags, accessories, skincare, makeup, home items, and more. We'll deliver it right to your doorstep with the convenient option of Cash on Delivery. Not sure about the sizing? You may return the item with our 30 days-free return policy. Purchase now and get the best deals and discounts for all the latest products only here at ZALORA Philippines. Shop today and you might just find incredible deals, discounts, promos, and exclusive ZALORA voucher codes for your favorite brands!

ARE ITEMS SOLD ON ZALORA ORIGINAL?

Yes, items sold on ZALORA are 100% authentic and original products from brands. ZALORA is an authorized retailer for all the brands available on their platform. They work directly with both luxury designer labels and popular high-street brands to bring genuine products to consumers in the Philippines. They follow stringent procurement processes directly from brand suppliers and vendors to verify the authenticity of items they list for sale on their e-commerce site. Buyers can shop with full confidence knowing the sneakers, clothes, bags, and accessories displayed on ZALORA abide by respective brands' quality standards and are covered by manufacturing defect warranties. Shopping policies also guarantee buyers can return items suspected to be inauthentic for a full refund. Ultimately, ZALORA strives to provide a trusted, transparent online shopping experience where customers access true original products from fashion’s biggest names.

DON'T LIKE TO WAIT? SIGN UP FOR ZALORA VIP TODAY FOR 365 UNLIMITED FREE SHIPPING

Hate paying the extra dime for delivery? Then sign up for ZALORA VIP to get unlimited free shipping for 365 days and other exclusive perks at only P500 per year! Extra advantages include priority access to sales, exclusive offers from ZALORA and partners, full rebates, and more. Shh, don’t tell anyone. Only ZALORA VIPs are entitled to exclusive deals, offers, and promotions.

The Hottest Picks Now On COS

Discover : COS Quilted Bag

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Shop by BrandsDiorCalvin KleinVersacePradaPUMATommy HilfigerCELINEBurberrySwarovskiCoachBVLGARIKate SpadeGUCCIKARL LAGERFELDTimberlandMichael KorsVejaPANDORABOSSTory BurchLONGCHAMPBALENCIAGAChanelALEXANDER MCQUEENBOTTEGA VENETAMIU MIUJo MaloneCole HaanOFF-WHITELouis VuittonTop BrandsALDOBirkenstockLacosteNikeSuperdryConverseCrocsNew BalanceRay-BanADIDASPUMACASIOGAPCLNMangoAmerican TouristerCotton On BodySwiss PoloCoachUnder ArmourCotton OnGUCCILondon RagHollisterVejaMARKS & SPENCERBOSSDr. MartensTory BurchFOREVER 21Show MoreTop SearchesBagsSportsClothesDiscount PricesAccessoriesShoesNew ProductsBeautyFormal DressBackpackShortsBlousePlus Size FashionWhite DressesSling BagsBohemian AttireLeatherCapsWomen's WatchesLong SleeveCasual DressesPerfumeHeelsDressCocktail DressesBraBlack DressSandalsMaxi DressJogger PantsShow MoreA company byOur locationsSecure paymentDelivery providersSecurity systemPCI Security StandardCardholder ProtectionEncrypted NetworkAs Asia's Online Fashion Destination, we create endless style possibilities through an ever-expanding range of products form the most coveted international and local brands, putting you at the centre of it all. With ZALORA, You Own Now.Customer serviceFAQSize GuideExchanges & ReturnsContact UsBuy Gift CardsProduct IndexBrandsAbout usWho We AreIntellectual PropertySell With UsCareersPromotionsInfluencer ProgramSustainability StrategyThread by ZALORAResponsible DisclosureThe Affiliate ProgramAdvertise with UsTrender Report 2021Terms & ConditionsPrivacy PolicyNPC Seal of RegistrationFind us onDownload our app nowAny question? Let us help you.About|Privacy|Terms of ServiceContact|Help|© 2012-2024 ZaloraTop BrandsView AllALDOBirkenstockLacosteNikeSuperdryConverseCrocsNew BalanceRay-BanADIDASPUMACASIOGAPCLNMangoAmerican TouristerCotton On BodySwiss PoloCoachUnder ArmourCotton OnGUCCILondon RagHollisterVejaMARKS & SPENCERBOSSDr. MartensTory BurchFOREVER 21Top SearchesShow MoreBagsSportsClothesDiscount PricesAccessoriesShoesNew ProductsBeautyFormal DressBackpackShortsBlousePlus Size FashionWhite DressesSling BagsBohemian AttireLeatherCapsWomen's WatchesLong SleeveCasual DressesPerfumeHeelsDressCocktail DressesBraBlack DressSandalsMaxi DressJogger PantsContactAboutLoginFAQTerms and ConditionsPrivacyHomeCategoryCartWishlistAcco

COS For Women 2024 | ZALORA Philippines

For Women 2024 | ZALORA PhilippinesLogin / RegisterWomenMenLuxurySportsKidsBeautyHome & LifestyleTop BrandsALDOBirkenstockLacosteNikeSuperdryConverseCrocsNew BalanceRay-BanADIDASPUMACASIOGAPCLNMangoAmerican TouristerCotton On BodySwiss PoloCoachUnder ArmourCotton OnGUCCILondon RagHollisterVejaMARKS & SPENCERBOSSDr. MartensTory BurchFOREVER 21Show MoreTop SearchesSportsAccessoriesNew ProductsBeautyBagsDiscount PricesShoesClothesLeatherJacketPerfumeCardiganFormal DressBlack DressPlus Size FashionMaxi DressSlippersBootsSandalsPastelsRash GuardCapsSneakersSwimwearCrossbody BagsWomen's WatchesHoodieCocktail DressesBeltCasual DressesShow MoreA company byOur locationsSecure paymentDelivery providersSecurity systemPCI Security StandardCardholder ProtectionEncrypted NetworkAs Asia's Online Fashion Destination, we create endless style possibilities through an ever-expanding range of products form the most coveted international and local brands, putting you at the centre of it all. With ZALORA, You Own Now.Customer serviceFAQSize GuideExchanges & ReturnsContact UsBuy Gift CardsProduct IndexBrandsAbout usWho We AreIntellectual PropertySell With UsCareersPromotionsInfluencer ProgramSustainability StrategyThread by ZALORAResponsible DisclosureThe Affiliate ProgramAdvertise with UsTrender Report 2021Terms & ConditionsPrivacy PolicyNPC Seal of RegistrationFind us onDownload our app nowAny question? Let us help you.About|Privacy|Terms of ServiceContact|Help|© 2012-2024 ZaloraTop BrandsView AllALDOBirkenstockLacosteNikeSuperdryConverseCrocsNew BalanceRay-BanADIDASPUMACASIOGAPCLNMangoAmerican TouristerCotton On BodySwiss PoloCoachUnder ArmourCotton OnGUCCILondon RagHollisterVejaMARKS & SPENCERBOSSDr. MartensTory BurchFOREVER 21Top SearchesShow MoreSportsAccessoriesNew ProductsBeautyBagsDiscount PricesShoesClothesLeatherJacketPerfumeCardiganFormal DressBlack DressPlus Size FashionMaxi DressSlippersBootsSandalsPastelsRash GuardCapsSneakersSwimwearCrossbody BagsWomen's WatchesHoodieCocktail DressesBeltCasual DressesContactAboutLoginFAQTerms and ConditionsPrivacyHomeCategoryCartWishlistAcco

COS - Contemporary Style, Designed To Last - Shop Online - WW

SPRING SUMMER 2024 COLLECTION | WOMEN MEN

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Spring Summer 2024

SS24 Womenswear

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COS Magazine: Jack O’Connell

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Spring Summer 2024: shop the collection

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COS magazine: read the new edition

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COS Full Circle

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Sustainability at COS: Better Looks Beyond

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SHOP WOMEN

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SPRING SUMMER 2024

women’s new arrivals

men's new arrivals

More to explore

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Women's Knitwear

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Enjoy 10% off your first order

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Womenswear Collection

SPRING SUMMER 2024 COLLECTION | WOMEN MEN

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The Quilted Bag

Shoes

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Trending Now

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Spring Summer 2024NEW

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COS Magazine: Jack O’Connell

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Spring Summer 2024: shop the collection

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Magazine

Read the latest

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Collections

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COS magazine: read the new edition

FM_MAGAZINE

Sustainability

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COS Full Circle

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COS Resell

Sustainability at COS: Better Looks Beyond

FM_SUSTAINABILITY

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THE SPRING SUMMER 2024 COLLECTION

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WIDE-LEG TAILORED TWILL TROUSERS

$135 USD

SPRING SUMMER 2024

Sold out

1226602001

1226602_group_001

WIDE-LEG TAILORED TWILL TROUSERS

USD

135.0

BLACK

Colours ()

BUTTON-DETAIL WOOL-BLEND WAISTCOAT

$225 USD

SPRING SUMMER 2024

Sold out

1218221001

1218221_group_001

BUTTON-DETAIL WOOL-BLEND WAISTCOAT

USD

225.0

BLACK

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PLEATED RACER-NECK MAXI DRESS

$175 USD

SPRING SUMMER 2024

Sold out

1216699002

1216699_group_002

PLEATED RACER-NECK MAXI DRESS

USD

175.0

RED

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CAVATELLI CLUTCH - LEATHER

$225 USD

SPRING SUMMER 2024

NEW

Sold out

1214221001

1214221_group_001

CAVATELLI CLUTCH - LEATHER

USD

225.0

BLACK

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MINI CAVATELLI CLUTCH - LEATHER

$135 USD

SPRING SUMMER 2024

Sold out

1214224001

1214224_group_001

MINI CAVATELLI CLUTCH - LEATHER

USD

135.0

BLACK

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QUILTED OVERSIZED CROSSBODY BAG - LEATHER

$490 USD

EXCLUDED FROM OFFER

Sold out

1200506001

1200506_group_001

QUILTED OVERSIZED CROSSBODY BAG - LEATHER

USD

490.0

BLACK

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ASYMMETRIC DRAPED MIDI DRESS

$175 USD

SPRING SUMMER 2024

Sold out

1221701002

1221701_group_002

ASYMMETRIC DRAPED MIDI DRESS

USD

175.0

RED

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STRIPED SPLIT-CUFF TROUSERS

$150 USD

SPRING SUMMER 2024

NEW

Sold out

1215867001

1215867_group_001

STRIPED SPLIT-CUFF TROUSERS

USD

150.0

RED / OFF-WHITE

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WIDE-LEG DRAWSTRING TROUSERS

$135 USD

SPRING SUMMER 2024

Sold out

1215961002

1215961_group_002

WIDE-LEG DRAWSTRING TROUSERS

USD

135.0

BEIGE

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HITCH OVERSIZED CROSSBODY - NYLON

$99 USD

SPRING SUMMER 2024

NEW

Sold out

1214247001

1214247_group_001

HITCH OVERSIZED CROSSBODY - NYLON

USD

99.0

BLACK

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CAVATELLI OVERSIZED CLUTCH - LEATHER

$390 USD

SPRING SUMMER 2024

NEW

Sold out

1231317001

1231317_group_001

CAVATELLI OVERSIZED CLUTCH - LEATHER

USD

390.0

RED

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EMBELLISHED CIRCLE-CUT MIDI SKIRT

$250 USD

SPRING SUMMER 2024

Sold out

1223365001

1223365_group_001

EMBELLISHED CIRCLE-CUT MIDI SKIRT

USD

250.0

WHITE

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BRODERIE ANGLAISE WESTERN SHIRT

$135 USD

SPRING SUMMER 2024

NEW

Sold out

1225546001

1225546_group_001

BRODERIE ANGLAISE WESTERN SHIRT

USD

135.0

WHITE

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DRAPED ASYMMETRIC MAXI DRESS

$225 USD

ECOVERO™

SPRING SUMMER 2024

Sold out

1221704001

1221704_group_001

DRAPED ASYMMETRIC MAXI DRESS

USD

225.0

WHITE

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FOLD MICRO TOTE - LEATHER

$150 USD

SPRING SUMMER 2024

Sold out

1223541001

1223541_group_001

FOLD MICRO TOTE - LEATHER

USD

150.0

BLACK

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WIDE-LEG TAILORED TWILL TROUSERS

$135 USD

SPRING SUMMER 2024

Sold out

1226602002

1226602_group_002

WIDE-LEG TAILORED TWILL TROUSERS

USD

135.0

CREAM

Colours ()

BROOCH-DETAIL WOOL-BLEND BLAZER

$250 USD

SPRING SUMMER 2024

NEW

Sold out

1218220001

1218220_group_001

BROOCH-DETAIL WOOL-BLEND BLAZER

USD

250.0

BLACK

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PLEATED RACER-NECK MAXI DRESS

$175 USD

SPRING SUMMER 2024

NEW

Sold out

1216699004

1216699_group_004

PLEATED RACER-NECK MAXI DRESS

USD

175.0

BLACK

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HOODED TRENCH COAT

$290 USD

SPRING SUMMER 2024

Sold out

1215478001

1215478_group_001

HOODED TRENCH COAT

USD

290.0

BEIGE

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FLARED WOOL-BLEND TROUSERS

$175 USD

SPRING SUMMER 2024

Sold out

1215966001

1215966_group_001

FLARED WOOL-BLEND TROUSERS

USD

175.0

BLACK

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FOLD MICRO TOTE - LEATHER

$150 USD

SPRING SUMMER 2024

NEW

Sold out

1223541002

1223541_group_002

FOLD MICRO TOTE - LEATHER

USD

150.0

OFF-WHITE

Colours ()

LONGLINE LINEN-BLEND WAISTCOAT

$135 USD

SPRING SUMMER 2024

NEW

Sold out

1208905002

1208905_group_002

LONGLINE LINEN-BLEND WAISTCOAT

USD

135.0

DUSTY PINK

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ORGANIC-SHAPED PENDANT NECKLACE

$69 USD

SPRING SUMMER 2024

Sold out

1224626001

1224626_group_001

ORGANIC-SHAPED PENDANT NECKLACE

USD

69.0

SILVER

Colours ()

OVERSIZED ORGANIC-SHAPED PENDANT NECKLACE

$99 USD

SPRING SUMMER 2024

Sold out

1224624001

1224624_group_001

OVERSIZED ORGANIC-SHAPED PENDANT NECKLACE

USD

99.0

SILVER

Colours ()

REVERSIBLE LEATHER BELT

$89 USD

SPRING SUMMER 2024

Sold out

1200472001

1200472_group_001

REVERSIBLE LEATHER BELT

USD

89.0

BLACK / LIGHT BEIGE

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PLEATED KNITTED TUNIC TOP

$135 USD

SPRING SUMMER 2024

Sold out

1219476001

1219476_group_001

PLEATED KNITTED TUNIC TOP

USD

135.0

BLACK

Colours ()

PLEATED KNITTED MIDI SKIRT

$150 USD

SPRING SUMMER 2024

Sold out

1214947001

1214947_group_001

PLEATED KNITTED MIDI SKIRT

USD

150.0

BLACK

Colours ()

ORGANIC-SHAPED PENDANT NECKLACE

$69 USD

SPRING SUMMER 2024

Sold out

1224626002

1224626_group_002

ORGANIC-SHAPED PENDANT NECKLACE

USD

69.0

GOLD

Colours ()

POINTED MESH SLINGBACK KITTEN HEELS

$225 USD

SPRING SUMMER 2024

Sold out

1216111001

1216111_group_001

POINTED MESH SLINGBACK KITTEN HEELS

USD

225.0

BLACK

Colours ()

TURN-UP DENIM TROUSERS

$150 USD

SPRING SUMMER 2024

Sold out

1214069001

1214069_group_001

TURN-UP DENIM TROUSERS

USD

150.0

OFF-WHITE

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MINI FOLDED CROSSBODY SHOPPER - LEATHER

$190 USD

SPRING SUMMER 2024

Sold out

1216127001

1216127_group_001

MINI FOLDED CROSSBODY SHOPPER - LEATHER

USD

190.0

BLACK

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RIBBED TANK TOP

$25 USD

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Over 95% of our collection is made from more sustainably sourced materials.

Rina Sawayama: resetting the status quo

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MathsMath ArticleCosine Function

Cosine Function

The cosine function (or cos function) in a triangle is the ratio of the adjacent side to that of the hypotenuse. The cosine function is one of the three main primary trigonometric functions and it is itself the complement of sine(co+sine). There are various topics that are included in the entire cos concept. Here, the main topics that are focussed include:

Cosine Definition

Cosine Formula

Cosine Table

Cosine Properties With Respect to the Quadrants

Cos Graph

Inverse Cosine (arccos)

Cosine Identities

Cos Calculus

Law of Cosines in Trigonometry

Additional Cos Values

Cosine Worksheet

Trigonometry Related Articles for Class 10

Trigonometry Related Articles for Class 11 and 12

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(Top)

1Notation

2Right-angled triangle definitions

3Radians versus degrees

4Unit-circle definitions

5Algebraic values

Toggle Algebraic values subsection

5.1Simple algebraic values

6In calculus

Toggle In calculus subsection

6.1Definition by differential equations

6.2Power series expansion

6.3Continued fraction expansion

6.4Partial fraction expansion

6.5Infinite product expansion

6.6Relationship to exponential function (Euler's formula)

6.7Definitions using functional equations

6.8In the complex plane

7Basic identities

Toggle Basic identities subsection

7.1Parity

7.2Periods

7.3Pythagorean identity

7.4Sum and difference formulas

7.5Derivatives and antiderivatives

8Inverse functions

9Applications

Toggle Applications subsection

9.1Angles and sides of a triangle

9.1.1Law of sines

9.1.2Law of cosines

9.1.3Law of tangents

9.1.4Law of cotangents

9.2Periodic functions

10History

11Etymology

12See also

13Notes

14References

15External links

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Trigonometric functions

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From Wikipedia, the free encyclopedia

Functions of an angle

"Logarithmic sine" redirects here. For the Clausen-related function, see log sine function.

"Logarithmic cosine" redirects here. For the Clausen-related function, see log cosine function.

Trigonometry

Outline

History

Usage

Functions (inverse)

Generalized trigonometry

Reference

Identities

Exact constants

Tables

Unit circle

Laws and theorems

Sines

Cosines

Tangents

Cotangents

Pythagorean theorem

Calculus

Trigonometric substitution

Integrals (inverse functions)

Derivatives

vte

Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their corresponding side lengths are proportional.

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions)[1][2] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions.

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.

Notation[edit]

Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation, for example sin(x). Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression

sin

⁡

{\displaystyle \sin x+y}

would typically be interpreted to mean

sin

⁡

(

)

{\displaystyle \sin(x)+y,}

so parentheses are required to express

sin

⁡

(

)

{\displaystyle \sin(x+y).}

A positive integer appearing as a superscript after the symbol of the function denotes exponentiation, not function composition. For example

sin

⁡

{\displaystyle \sin ^{2}x}

and

sin

⁡

(

)

{\displaystyle \sin ^{2}(x)}

denote

sin

⁡

(

)

⋅

sin

⁡

(

)

{\displaystyle \sin(x)\cdot \sin(x),}

not

sin

⁡

(

sin

⁡

)

{\displaystyle \sin(\sin x).}

This differs from the (historically later) general functional notation in which

(

)

(

∘

)

(

)

(

)

{\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}

However, the exponent

−

{\displaystyle {-1}}

is commonly used to denote the inverse function, not the reciprocal. For example

sin

−

⁡

{\displaystyle \sin ^{-1}x}

and

sin

−

⁡

(

)

{\displaystyle \sin ^{-1}(x)}

denote the inverse trigonometric function alternatively written

arcsin

⁡

{\displaystyle \arcsin x\colon }

The equation

sin

−

⁡

{\displaystyle \theta =\sin ^{-1}x}

implies

sin

⁡

{\displaystyle \sin \theta =x,}

not

⋅

sin

⁡

{\displaystyle \theta \cdot \sin x=1.}

In this case, the superscript could be considered as denoting a composed or iterated function, but negative superscripts other than

−

{\displaystyle {-1}}

are not in common use.

Right-angled triangle definitions[edit]

In this right triangle, denoting the measure of angle BAC as A: sin A = a/c; cos A = b/c; tan A = a/b.

Plot of the six trigonometric functions, the unit circle, and a line for the angle θ = 0.7 radians. The points labelled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point. Sin(θ), Tan(θ), and 1 are the heights to the line starting from the x-axis, while Cos(θ), 1, and Cot(θ) are lengths along the x-axis starting from the origin.

If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ. Thus these six ratios define six functions of θ, which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ, and adjacent represents the side between the angle θ and the right angle.[3][4]

sine

sin

⁡

{\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}}

cosecant

csc

⁡

{\displaystyle \csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}}

cosine

cos

⁡

{\displaystyle \cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}}

secant

sec

⁡

{\displaystyle \sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}}

tangent

tan

⁡

{\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}

cotangent

cot

⁡

{\displaystyle \cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}}

Various mnemonics can be used to remember these definitions.

In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π/2 radians. Therefore

sin

⁡

(

)

{\displaystyle \sin(\theta )}

and

cos

⁡

(

∘

−

)

{\displaystyle \cos(90^{\circ }-\theta )}

represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.

Top: Trigonometric function sin θ for selected angles θ, π − θ, π + θ, and 2π − θ in the four quadrants.Bottom: Graph of sine function versus angle. Angles from the top panel are identified.

Summary of relationships between trigonometric functions[5]

Function

Description

Relationship

using radians

using degrees

sine

opposite/hypotenuse

sin

⁡

cos

⁡

(

−

)

csc

⁡

{\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}}

sin

⁡

cos

⁡

(

∘

−

)

csc

⁡

{\displaystyle \sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}}

cosine

adjacent/hypotenuse

cos

⁡

sin

⁡

(

−

)

sec

⁡

{\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}

cos

⁡

sin

⁡

(

∘

−

)

sec

⁡

{\displaystyle \cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,}

tangent

opposite/adjacent

tan

⁡

sin

⁡

cos

⁡

cot

⁡

(

−

)

cot

⁡

{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}}

tan

⁡

sin

⁡

cos

⁡

cot

⁡

(

∘

−

)

cot

⁡

{\displaystyle \tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}}

cotangent

adjacent/opposite

cot

⁡

cos

⁡

sin

⁡

tan

⁡

(

−

)

tan

⁡

{\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}}

cot

⁡

cos

⁡

sin

⁡

tan

⁡

(

∘

−

)

tan

⁡

{\displaystyle \cot x={\frac {\cos x}{\sin x}}=\tan \left(90^{\circ }-x\right)={\frac {1}{\tan x}}}

secant

hypotenuse/adjacent

sec

⁡

csc

⁡

(

−

)

cos

⁡

{\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}}

sec

⁡

csc

⁡

(

∘

−

)

cos

⁡

{\displaystyle \sec x=\csc \left(90^{\circ }-x\right)={\frac {1}{\cos x}}}

cosecant

hypotenuse/opposite

csc

⁡

sec

⁡

(

−

)

sin

⁡

{\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}}

csc

⁡

sec

⁡

(

∘

−

)

sin

⁡

{\displaystyle \csc x=\sec \left(90^{\circ }-x\right)={\frac {1}{\sin x}}}

Radians versus degrees[edit]

In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient. One common unit is degrees, in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics).

However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function, via power series,[6] or as solutions to differential equations given particular initial values[7] (see below), without reference to any geometric notions. The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle given in radians.[6] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions.[8] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.

When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2π (≈ 6.28) rad. For real number x, the notations sin x, cos x, etc. refer to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g., sin x°, cos x°, etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/π)°, so that, for example, sin π = sin 180° when we take x = π. In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π/180 ≈ 0.0175.

Unit-circle definitions[edit]

All of the trigonometric functions of the angle θ (theta) can be constructed geometrically in terms of a unit circle centered at O.

Sine function on unit circle (top) and its graph (bottom)

In this illustration, the six trigonometric functions of an arbitrary angle θ are represented as Cartesian coordinates of points related to the unit circle. The ordinates of A, B and D are sin θ, tan θ and csc θ, respectively, while the abscissas of A, C and E are cos θ, cot θ and sec θ, respectively.

Signs of trigonometric functions in each quadrant. The mnemonic "all science teachers (are) crazy" indicates when sine, cosine, and tangent are positive from quadrants I to IV.[9]

The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle, which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and

{\textstyle {\frac {\pi }{2}}}

radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.

Let

{\displaystyle {\mathcal {L}}}

be the ray obtained by rotating by an angle θ the positive half of the x-axis (counterclockwise rotation for

{\displaystyle \theta >0,}

and clockwise rotation for

{\displaystyle \theta <0}

). This ray intersects the unit circle at the point

(

)

{\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).}

The ray

{\displaystyle {\mathcal {L}},}

extended to a line if necessary, intersects the line of equation

{\displaystyle x=1}

at point

(

)

{\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),}

and the line of equation

{\displaystyle y=1}

at point

(

)

{\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).}

The tangent line to the unit circle at the point A, is perpendicular to

{\displaystyle {\mathcal {L}},}

and intersects the y- and x-axes at points

(

)

{\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })}

and

(

)

{\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).}

The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.

The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A. That is,

cos

⁡

{\displaystyle \cos \theta =x_{\mathrm {A} }\quad }

and

sin

⁡

{\displaystyle \quad \sin \theta =y_{\mathrm {A} }.}

[10]

In the range

≤

{\displaystyle 0\leq \theta \leq \pi /2}

, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse. And since the equation

{\displaystyle x^{2}+y^{2}=1}

holds for all points

(

)

{\displaystyle \mathrm {P} =(x,y)}

on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity.

cos

⁡

sin

⁡

{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}

The other trigonometric functions can be found along the unit circle as

tan

⁡

{\displaystyle \tan \theta =y_{\mathrm {B} }\quad }

and

cot

⁡

{\displaystyle \quad \cot \theta =x_{\mathrm {C} },}

csc

⁡

{\displaystyle \csc \theta \ =y_{\mathrm {D} }\quad }

and

sec

⁡

{\displaystyle \quad \sec \theta =x_{\mathrm {E} }.}

By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is

tan

⁡

sin

⁡

cos

⁡

cot

⁡

cos

⁡

sin

⁡

sec

⁡

cos

⁡

csc

⁡

sin

⁡

{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }},\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }},\quad \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }}.}

Trigonometric functions: Sine, Cosine, Tangent, Cosecant (dotted), Secant (dotted), Cotangent (dotted) – animation

Since a rotation of an angle of

{\displaystyle \pm 2\pi }

does not change the position or size of a shape, the points A, B, C, D, and E are the same for two angles whose difference is an integer multiple of

{\displaystyle 2\pi }

. Thus trigonometric functions are periodic functions with period

{\displaystyle 2\pi }

. That is, the equalities

sin

⁡

sin

⁡

(

)

{\displaystyle \sin \theta =\sin \left(\theta +2k\pi \right)\quad }

and

cos

⁡

cos

⁡

(

)

{\displaystyle \quad \cos \theta =\cos \left(\theta +2k\pi \right)}

hold for any angle θ and any integer k. The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that

{\displaystyle 2\pi }

is the smallest value for which they are periodic (i.e.,

{\displaystyle 2\pi }

is the fundamental period of these functions). However, after a rotation by an angle

{\displaystyle \pi }

, the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of

{\displaystyle \pi }

. That is, the equalities

tan

⁡

tan

⁡

(

)

{\displaystyle \tan \theta =\tan(\theta +k\pi )\quad }

and

cot

⁡

cot

⁡

(

)

{\displaystyle \quad \cot \theta =\cot(\theta +k\pi )}

hold for any angle θ and any integer k.

Algebraic values[edit]

The unit circle, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.

The algebraic expressions for the most important angles are as follows:

sin

⁡

sin

⁡

∘

{\displaystyle \sin 0=\sin 0^{\circ }\quad ={\frac {\sqrt {0}}{2}}=0}

(zero angle)

sin

⁡

sin

⁡

∘

{\displaystyle \sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {\sqrt {1}}{2}}={\frac {1}{2}}}

sin

⁡

sin

⁡

∘

{\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}

sin

⁡

sin

⁡

∘

{\displaystyle \sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}}

sin

⁡

sin

⁡

∘

{\displaystyle \sin {\frac {\pi }{2}}=\sin 90^{\circ }={\frac {\sqrt {4}}{2}}=1}

(right angle)

Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.[11]

Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.

For an angle which, measured in degrees, is a multiple of three, the exact trigonometric values of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by ruler and compass.

For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the cube root of a non-real complex number. Galois theory allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.

For an angle which, expressed in degrees, is a rational number, the sine and the cosine are algebraic numbers, which may be expressed in terms of nth roots. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic.

For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine are transcendental numbers. This is a corollary of Baker's theorem, proved in 1966.

Simple algebraic values[edit]

Main article: Exact trigonometric values § Common angles

The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.

Angle, θ, in

sin

⁡

(

)

{\displaystyle \sin(\theta )}

cos

⁡

(

)

{\displaystyle \cos(\theta )}

tan

⁡

(

)

{\displaystyle \tan(\theta )}

radians

degrees

{\displaystyle 0}

∘

{\displaystyle 0^{\circ }}

{\displaystyle 0}

{\displaystyle 1}

{\displaystyle 0}

{\displaystyle {\frac {\pi }{12}}}

∘

{\displaystyle 15^{\circ }}

−

{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}

{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}

−

{\displaystyle 2-{\sqrt {3}}}

{\displaystyle {\frac {\pi }{6}}}

∘

{\displaystyle 30^{\circ }}

{\displaystyle {\frac {1}{2}}}

{\displaystyle {\frac {\sqrt {3}}{2}}}

{\displaystyle {\frac {\sqrt {3}}{3}}}

{\displaystyle {\frac {\pi }{4}}}

∘

{\displaystyle 45^{\circ }}

{\displaystyle {\frac {\sqrt {2}}{2}}}

[a]

{\displaystyle 1}

{\displaystyle {\frac {\pi }{3}}}

∘

{\displaystyle 60^{\circ }}

{\displaystyle {\frac {\sqrt {3}}{2}}}

{\displaystyle {\frac {1}{2}}}

{\displaystyle {\sqrt {3}}}

{\displaystyle {\frac {5\pi }{12}}}

∘

{\displaystyle 75^{\circ }}

{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}

−

{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}

{\displaystyle 2+{\sqrt {3}}}

{\displaystyle {\frac {\pi }{2}}}

∘

{\displaystyle 90^{\circ }}

{\displaystyle 1}

{\displaystyle 0}

Undefined

In calculus[edit]

Graphs of sine, cosine and tangent

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

Animation for the approximation of cosine via Taylor polynomials.

cos

⁡

(

)

{\displaystyle \cos(x)}

together with the first Taylor polynomials

(

)

∑

(

−

)

(

)

{\displaystyle p_{n}(x)=\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}}

The modern trend in mathematics is to build geometry from calculus rather than the converse.[citation needed] Therefore, except at a very elementary level, trigonometric functions are defined using the methods of calculus.

Trigonometric functions are differentiable and analytic at every point where they are defined; that is, everywhere for the sine and the cosine, and, for the tangent, everywhere except at π/2 + kπ for every integer k.

The trigonometric function are periodic functions, and their primitive period is 2π for the sine and the cosine, and π for the tangent, which is increasing in each open interval (π/2 + kπ, π/2 + (k + 1)π). At each end point of these intervals, the tangent function has a vertical asymptote.

In calculus, there are two equivalent definitions of trigonometric functions, either using power series or differential equations. These definitions are equivalent, as starting from one of them, it is easy to retrieve the other as a property. However the definition through differential equations is somehow more natural, since, for example, the choice of the coefficients of the power series may appear as quite arbitrary, and the Pythagorean identity is much easier to deduce from the differential equations.

Definition by differential equations[edit]

Sine and cosine can be defined as the unique solution to the initial value problem:

sin

⁡

cos

⁡

cos

⁡

−

sin

⁡

sin

⁡

(

)

cos

⁡

(

)

{\displaystyle {\frac {d}{dx}}\sin x=\cos x,\ {\frac {d}{dx}}\cos x=-\sin x,\ \sin(0)=0,\ \cos(0)=1.}

Differentiating again,

sin

⁡

cos

⁡

−

sin

⁡

{\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x}

and

cos

⁡

−

sin

⁡

−

cos

⁡

{\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x}

, so both sine and cosine are solutions of the same ordinary differential equation

″

{\displaystyle y''+y=0\,.}

Sine is the unique solution with y(0) = 0 and y′(0) = 1; cosine is the unique solution with y(0) = 1 and y′(0) = 0.

Applying the quotient rule to the tangent

tan

⁡

sin

⁡

cos

⁡

{\displaystyle \tan x=\sin x/\cos x}

tan

⁡

cos

⁡

sin

⁡

cos

⁡

tan

⁡

{\displaystyle {\frac {d}{dx}}\tan x={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}=1+\tan ^{2}x\,,}

so the tangent function satisfies the ordinary differential equation

′

{\displaystyle y'=1+y^{2}\,.}

It is the unique solution with y(0) = 0.

Power series expansion[edit]

Applying the differential equations to power series with indeterminate coefficients, one may deduce recurrence relations for the coefficients of the Taylor series of the sine and cosine functions. These recurrence relations are easy to solve, and give the series expansions[12]

sin

⁡

−

⋯

∑

∞

(

−

)

(

)

cos

⁡

−

⋯

∑

∞

(

−

)

(

)

{\displaystyle {\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}.\end{aligned}}}

The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane.

Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles. Here, the poles are the numbers of the form

(

)

{\textstyle (2k+1){\frac {\pi }{2}}}

for the tangent and the secant, or

{\displaystyle k\pi }

for the cotangent and the cosecant, where k is an arbitrary integer.

Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets.[13]

More precisely, defining

Un, the nth up/down number,

Bn, the nth Bernoulli number, and

En, is the nth Euler number,

one has the following series expansions:[14]

tan

⁡

∑

∞

(

)

∑

∞

(

−

)

−

(

−

)

(

)

−

315

⋯

for

{\displaystyle {\begin{aligned}\tan x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}}{(2n+1)!}}x^{2n+1}\\[8mu]&{}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}\left(2^{2n}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&{}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}

csc

⁡

∑

∞

(

−

)

(

−

)

(

)

−

360

15120

⋯

for

{\displaystyle {\begin{aligned}\csc x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}+{\frac {1}{6}}x+{\frac {7}{360}}x^{3}+{\frac {31}{15120}}x^{5}+\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}

sec

⁡

∑

∞

(

)

∑

∞

(

−

)

(

)

720

⋯

for

{\displaystyle {\begin{aligned}\sec x&=\sum _{n=0}^{\infty }{\frac {U_{2n}}{(2n)!}}x^{2n}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}\\[5mu]&=1+{\frac {1}{2}}x^{2}+{\frac {5}{24}}x^{4}+{\frac {61}{720}}x^{6}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}

cot

⁡

∑

∞

(

−

)

(

)

−

945

−

⋯

for

{\displaystyle {\begin{aligned}\cot x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}

Continued fraction expansion[edit]

The following expansions are valid in the whole complex plane:

sin

⁡

⋅

−

⋅

−

⋅

−

⋱

{\displaystyle \sin x={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}}

cos

⁡

⋅

−

⋅

−

⋅

−

⋱

{\displaystyle \cos x={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}}

tan

⁡

−

⋱

−

⋱

{\displaystyle \tan x={\cfrac {x}{1-{\cfrac {x^{2}}{3-{\cfrac {x^{2}}{5-{\cfrac {x^{2}}{7-\ddots }}}}}}}}={\cfrac {1}{{\cfrac {1}{x}}-{\cfrac {1}{{\cfrac {3}{x}}-{\cfrac {1}{{\cfrac {5}{x}}-{\cfrac {1}{{\cfrac {7}{x}}-\ddots }}}}}}}}}

The last one was used in the historically first proof that π is irrational.[15]

Partial fraction expansion[edit]

There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match:[16]

cot

⁡

lim

→

∞

∑

−

{\displaystyle \pi \cot \pi x=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.}

This identity can be proved with the Herglotz trick.[17]

Combining the (–n)th with the nth term lead to absolutely convergent series:

cot

⁡

∑

∞

−

{\displaystyle \pi \cot \pi x={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {1}{x^{2}-n^{2}}}.}

Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:

csc

⁡

∑

−

∞

(

−

)

∑

∞

(

−

)

−

{\displaystyle \pi \csc \pi x=\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{x+n}}={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{x^{2}-n^{2}}},}

csc

⁡

∑

−

∞

(

)

{\displaystyle \pi ^{2}\csc ^{2}\pi x=\sum _{n=-\infty }^{\infty }{\frac {1}{(x+n)^{2}}},}

sec

⁡

∑

∞

(

−

)

(

)

(

)

−

{\displaystyle \pi \sec \pi x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n+1)}{(n+{\tfrac {1}{2}})^{2}-x^{2}}},}

tan

⁡

∑

∞

(

)

−

{\displaystyle \pi \tan \pi x=2x\sum _{n=0}^{\infty }{\frac {1}{(n+{\tfrac {1}{2}})^{2}-x^{2}}}.}

Infinite product expansion[edit]

The following infinite product for the sine is of great importance in complex analysis:

sin

⁡

∏

∞

(

−

)

∈

{\displaystyle \sin z=z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}

For the proof of this expansion, see Sine. From this, it can be deduced that

cos

⁡

∏

∞

(

−

(

−

)

∈

{\displaystyle \cos z=\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{(n-1/2)^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}

Relationship to exponential function (Euler's formula)[edit]

cos

⁡

(

)

{\displaystyle \cos(\theta )}

and

sin

⁡

(

)

{\displaystyle \sin(\theta )}

are the real and imaginary part of

{\displaystyle e^{i\theta }}

respectively.

Euler's formula relates sine and cosine to the exponential function:

cos

⁡

sin

⁡

{\displaystyle e^{ix}=\cos x+i\sin x.}

This formula is commonly considered for real values of x, but it remains true for all complex values.

Proof: Let

(

)

cos

⁡

sin

⁡

{\displaystyle f_{1}(x)=\cos x+i\sin x,}

and

(

)

{\displaystyle f_{2}(x)=e^{ix}.}

One has

(

)

(

)

{\displaystyle df_{j}(x)/dx=if_{j}(x)}

for j = 1, 2. The quotient rule implies thus that

(

)

(

)

{\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0}

. Therefore,

(

)

(

)

{\displaystyle f_{1}(x)/f_{2}(x)}

is a constant function, which equals 1, as

(

)

(

)

{\displaystyle f_{1}(0)=f_{2}(0)=1.}

This proves the formula.

One has

cos

⁡

sin

⁡

−

cos

⁡

−

sin

⁡

{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\[5pt]e^{-ix}&=\cos x-i\sin x.\end{aligned}}}

Solving this linear system in sine and cosine, one can express them in terms of the exponential function:

sin

⁡

−

cos

⁡

−

{\displaystyle {\begin{aligned}\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\[5pt]\cos x&={\frac {e^{ix}+e^{-ix}}{2}}.\end{aligned}}}

When x is real, this may be rewritten as

cos

⁡

(

)

sin

⁡

(

)

{\displaystyle \cos x=\operatorname {Re} \left(e^{ix}\right),\qquad \sin x=\operatorname {Im} \left(e^{ix}\right).}

Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity

{\displaystyle e^{a+b}=e^{a}e^{b}}

for simplifying the result.

Definitions using functional equations[edit]

One can also define the trigonometric functions using various functional equations.

For example,[18] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula

cos

⁡

(

−

)

cos

⁡

cos

⁡

sin

⁡

sin

⁡

{\displaystyle \cos(x-y)=\cos x\cos y+\sin x\sin y\,}

and the added condition

cos

⁡

sin

⁡

for

{\displaystyle 0

In the complex plane[edit]

The sine and cosine of a complex number

{\displaystyle z=x+iy}

can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:

sin

⁡

sin

⁡

cosh

⁡

cos

⁡

sinh

⁡

cos

⁡

cos

⁡

cosh

⁡

−

sin

⁡

sinh

⁡

{\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}}

By taking advantage of domain coloring, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of

{\displaystyle z}

becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.

Trigonometric functions in the complex plane

sin

⁡

{\displaystyle \sin z\,}

cos

⁡

{\displaystyle \cos z\,}

tan

⁡

{\displaystyle \tan z\,}

cot

⁡

{\displaystyle \cot z\,}

sec

⁡

{\displaystyle \sec z\,}

csc

⁡

{\displaystyle \csc z\,}

Basic identities[edit]

Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π/2], see Proofs of trigonometric identities). For non-geometrical proofs using only tools of calculus, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.

Parity[edit]

The cosine and the secant are even functions; the other trigonometric functions are odd functions. That is:

sin

⁡

(

−

)

−

sin

⁡

cos

⁡

(

−

)

cos

⁡

tan

⁡

(

−

)

−

tan

⁡

cot

⁡

(

−

)

−

cot

⁡

csc

⁡

(

−

)

−

csc

⁡

sec

⁡

(

−

)

sec

⁡

{\displaystyle {\begin{aligned}\sin(-x)&=-\sin x\\\cos(-x)&=\cos x\\\tan(-x)&=-\tan x\\\cot(-x)&=-\cot x\\\csc(-x)&=-\csc x\\\sec(-x)&=\sec x.\end{aligned}}}

Periods[edit]

All trigonometric functions are periodic functions of period 2π. This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k, one has

sin

⁡

(

)

sin

⁡

cos

⁡

(

)

cos

⁡

tan

⁡

(

)

tan

⁡

cot

⁡

(

)

cot

⁡

csc

⁡

(

)

csc

⁡

sec

⁡

(

)

sec

⁡

{\displaystyle {\begin{array}{lrl}\sin(x+&2k\pi )&=\sin x\\\cos(x+&2k\pi )&=\cos x\\\tan(x+&k\pi )&=\tan x\\\cot(x+&k\pi )&=\cot x\\\csc(x+&2k\pi )&=\csc x\\\sec(x+&2k\pi )&=\sec x.\end{array}}}

Pythagorean identity[edit]

The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is

sin

⁡

cos

⁡

{\displaystyle \sin ^{2}x+\cos ^{2}x=1}

Dividing through by either

cos

⁡

{\displaystyle \cos ^{2}x}

sin

⁡

{\displaystyle \sin ^{2}x}

gives

tan

⁡

sec

⁡

{\displaystyle \tan ^{2}x+1=\sec ^{2}x}

and

cot

⁡

csc

⁡

{\displaystyle 1+\cot ^{2}x=\csc ^{2}x}

Sum and difference formulas[edit]

The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula.

Sum

sin

⁡

(

)

sin

⁡

cos

⁡

cos

⁡

sin

⁡

cos

⁡

(

)

cos

⁡

cos

⁡

−

sin

⁡

sin

⁡

tan

⁡

(

)

tan

⁡

tan

⁡

−

tan

⁡

tan

⁡

{\displaystyle {\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\[5mu]\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\[5mu]\tan(x+y)&={\frac {\tan x+\tan y}{1-\tan x\tan y}}.\end{aligned}}}

Difference

sin

⁡

(

−

)

sin

⁡

cos

⁡

−

cos

⁡

sin

⁡

cos

⁡

(

−

)

cos

⁡

cos

⁡

sin

⁡

sin

⁡

tan

⁡

(

−

)

tan

⁡

−

tan

⁡

tan

⁡

tan

⁡

{\displaystyle {\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\[5mu]\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\[5mu]\tan(x-y)&={\frac {\tan x-\tan y}{1+\tan x\tan y}}.\end{aligned}}}

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.

sin

⁡

sin

⁡

cos

⁡

tan

⁡

tan

⁡

cos

⁡

cos

⁡

−

sin

⁡

cos

⁡

−

sin

⁡

−

tan

⁡

tan

⁡

tan

⁡

tan

⁡

−

tan

⁡

{\displaystyle {\begin{aligned}\sin 2x&=2\sin x\cos x={\frac {2\tan x}{1+\tan ^{2}x}},\\[5mu]\cos 2x&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}},\\[5mu]\tan 2x&={\frac {2\tan x}{1-\tan ^{2}x}}.\end{aligned}}}

These identities can be used to derive the product-to-sum identities.

By setting

tan

⁡

{\displaystyle t=\tan {\tfrac {1}{2}}\theta ,}

all trigonometric functions of

{\displaystyle \theta }

can be expressed as rational fractions of

{\displaystyle t}

sin

⁡

cos

⁡

−

tan

⁡

−

{\displaystyle {\begin{aligned}\sin \theta &={\frac {2t}{1+t^{2}}},\\[5mu]\cos \theta &={\frac {1-t^{2}}{1+t^{2}}},\\[5mu]\tan \theta &={\frac {2t}{1-t^{2}}}.\end{aligned}}}

Together with

{\displaystyle d\theta ={\frac {2}{1+t^{2}}}\,dt,}

this is the tangent half-angle substitution, which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.

Derivatives and antiderivatives[edit]

The derivatives of trigonometric functions result from those of sine and cosine by applying quotient rule. The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration.

(

)

{\displaystyle f(x)}

′

(

)

{\displaystyle f'(x)}

∫

(

)

{\textstyle \int f(x)\,dx}

sin

⁡

{\displaystyle \sin x}

cos

⁡

{\displaystyle \cos x}

−

cos

⁡

{\displaystyle -\cos x+C}

cos

⁡

{\displaystyle \cos x}

−

sin

⁡

{\displaystyle -\sin x}

sin

⁡

{\displaystyle \sin x+C}

tan

⁡

{\displaystyle \tan x}

sec

⁡

{\displaystyle \sec ^{2}x}

⁡

sec

⁡

{\displaystyle \ln \left|\sec x\right|+C}

csc

⁡

{\displaystyle \csc x}

−

csc

⁡

cot

⁡

{\displaystyle -\csc x\cot x}

⁡

csc

⁡

−

cot

⁡

{\displaystyle \ln \left|\csc x-\cot x\right|+C}

sec

⁡

{\displaystyle \sec x}

sec

⁡

tan

⁡

{\displaystyle \sec x\tan x}

⁡

sec

⁡

tan

⁡

{\displaystyle \ln \left|\sec x+\tan x\right|+C}

cot

⁡

{\displaystyle \cot x}

−

csc

⁡

{\displaystyle -\csc ^{2}x}

⁡

sin

⁡

{\displaystyle \ln \left|\sin x\right|+C}

Note: For

{\displaystyle 0

the integral of

csc

⁡

{\displaystyle \csc x}

can also be written as

−

arsinh

⁡

(

cot

⁡

)

{\displaystyle -\operatorname {arsinh} (\cot x),}

and for the integral of

sec

⁡

{\displaystyle \sec x}

for

−

{\displaystyle -\pi /2

arsinh

⁡

(

tan

⁡

)

{\displaystyle \operatorname {arsinh} (\tan x),}

where

arsinh

{\displaystyle \operatorname {arsinh} }

is the inverse hyperbolic sine.

Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:

cos

⁡

sin

⁡

(

−

)

−

cos

⁡

(

−

)

−

sin

⁡

csc

⁡

sec

⁡

(

−

)

−

sec

⁡

(

−

)

tan

⁡

(

−

)

−

csc

⁡

cot

⁡

cot

⁡

tan

⁡

(

−

)

−

sec

⁡

(

−

)

−

csc

⁡

{\displaystyle {\begin{aligned}{\frac {d\cos x}{dx}}&={\frac {d}{dx}}\sin(\pi /2-x)=-\cos(\pi /2-x)=-\sin x\,,\\{\frac {d\csc x}{dx}}&={\frac {d}{dx}}\sec(\pi /2-x)=-\sec(\pi /2-x)\tan(\pi /2-x)=-\csc x\cot x\,,\\{\frac {d\cot x}{dx}}&={\frac {d}{dx}}\tan(\pi /2-x)=-\sec ^{2}(\pi /2-x)=-\csc ^{2}x\,.\end{aligned}}}

Inverse functions[edit]

Main article: Inverse trigonometric functions

The trigonometric functions are periodic, and hence not injective, so strictly speaking, they do not have an inverse function. However, on each interval on which a trigonometric function is monotonic, one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values, is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.

Function

Definition

Domain

Set of principal values

arcsin

⁡

{\displaystyle y=\arcsin x}

sin

⁡

{\displaystyle \sin y=x}

−

≤

{\displaystyle -1\leq x\leq 1}

−

≤

{\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}

arccos

⁡

{\displaystyle y=\arccos x}

cos

⁡

{\displaystyle \cos y=x}

−

≤

{\displaystyle -1\leq x\leq 1}

≤

{\textstyle 0\leq y\leq \pi }

arctan

⁡

{\displaystyle y=\arctan x}

tan

⁡

{\displaystyle \tan y=x}

−

∞

{\displaystyle -\infty

−

{\textstyle -{\frac {\pi }{2}}

arccot

⁡

{\displaystyle y=\operatorname {arccot} x}

cot

⁡

{\displaystyle \cot y=x}

−

∞

{\displaystyle -\infty

{\textstyle 0

arcsec

⁡

{\displaystyle y=\operatorname {arcsec} x}

sec

⁡

{\displaystyle \sec y=x}

−

{\displaystyle x<-1{\text{ or }}x>1}

≤

≠

{\textstyle 0\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}}

arccsc

⁡

{\displaystyle y=\operatorname {arccsc} x}

csc

⁡

{\displaystyle \csc y=x}

−

{\displaystyle x<-1{\text{ or }}x>1}

−

≤

≠

{\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 0}

The notations sin−1, cos−1, etc. are often used for arcsin and arccos, etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with "arcsecond".

Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms.

Applications[edit]

Main article: Uses of trigonometry

Angles and sides of a triangle[edit]

In this section A, B, C denote the three (interior) angles of a triangle, and a, b, c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.

Law of sines[edit]

Main article: Law of sines

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:

sin

⁡

sin

⁡

sin

⁡

{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},}

where Δ is the area of the triangle,

or, equivalently,

sin

⁡

sin

⁡

sin

⁡

{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}

where R is the triangle's circumradius.

It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

Law of cosines[edit]

Main article: Law of cosines

The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem:

−

cos

⁡

{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,}

or equivalently,

cos

⁡

−

{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}

In this formula the angle at C is opposite to the side c. This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem.

The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.

Law of tangents[edit]

Main article: Law of tangents

The law of tangents says that:

tan

⁡

−

tan

⁡

−

{\displaystyle {\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}}

Law of cotangents[edit]

Main article: Law of cotangents

If s is the triangle's semiperimeter, (a + b + c)/2, and r is the radius of the triangle's incircle, then rs is the triangle's area. Therefore Heron's formula implies that:

(

−

)

(

−

)

(

−

)

{\displaystyle r={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}}

The law of cotangents says that:[19]

cot

⁡

−

{\displaystyle \cot {\frac {A}{2}}={\frac {s-a}{r}}}

It follows that

cot

⁡

−

cot

⁡

−

cot

⁡

−

{\displaystyle {\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{r}}.}

Periodic functions[edit]

A Lissajous curve, a figure formed with a trigonometry-based function.

An animation of the additive synthesis of a square wave with an increasing number of harmonics

Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental (k = 1) have additional nodes. The oscillation seen about the sawtooth when k is large is called the Gibbs phenomenon

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.

Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.[20]

Under rather general conditions, a periodic function f (x) can be expressed as a sum of sine waves or cosine waves in a Fourier series.[21] Denoting the sine or cosine basis functions by φk, the expansion of the periodic function f (t) takes the form:

(

)

∑

∞

(

)

{\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}

For example, the square wave can be written as the Fourier series

square

(

)

∑

∞

sin

⁡

(

−

)

−

{\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}

In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.

History[edit]

Main article: History of trigonometry

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) can be traced back to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.[22] (See Aryabhata's sine table.)

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles.[23] With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Persian and Arab mathematicians, including the cosine, tangent, cotangent, secant and cosecant.[23] Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Circa 830, Habash al-Hasib al-Marwazi discovered the cotangent, and produced tables of tangents and cotangents.[24][25] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.[25] The trigonometric functions were later studied by mathematicians including Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus Otho.

Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series.[26] (See Madhava series and Madhava's sine table.)

The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.[27]

The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583).[28]

The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin, cos, and tan in his book Trigonométrie.[29]

In a paper published in 1682, Gottfried Leibniz proved that sin x is not an algebraic function of x.[30] Though introduced as ratios of sides of a right triangle, and thus appearing to be rational functions, Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series. He presented "Euler's formula", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.).[22]

A few functions were common historically, but are now seldom used, such as the chord, the versine (which appeared in the earliest tables[22]), the coversine, the haversine,[31] the exsecant and the excosecant. The list of trigonometric identities shows more relations between these functions.

crd(θ) = 2 sin(θ/2)

versin(θ) = 1 − cos(θ) = 2 sin2(θ/2)

coversin(θ) = 1 − sin(θ) = versin(π/2 − θ)

haversin(θ) = 1/2versin(θ) = sin2(θ/2)

exsec(θ) = sec(θ) − 1

excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1

Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent.[32][33][34][35]

Etymology[edit]

Main article: History of trigonometry § Etymology

The word sine derives[36] from Latin sinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin.[37]

The choice was based on a misreading of the Arabic written form j-y-b (جيب), which itself originated as a transliteration from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string".[38]

The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans—"cutting"—since the line cuts the circle.[39]

The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.[40][41]

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