Trigonometric Identities Chart Reciprocal Identities Quotient Identities ...

Precalculus HS Mathematics Unit: 05 Lesson: 02

T r i g o n o m e t r i c Identities C h a r t Reciprocal Identities sinx = cosx =

cscx =

cscx 1

secx =

secx 1

tanx =

Sum and Difference Identities 1

1

cotx =

cotx

sinx 1

sin(>A + 8 )

sin/4 c o s 8 + c o s / A s i n B

sin(>A-S)

sin/AcosS-cos/AsinB

cosx 1

cos(/A + S ) = : c o s i4 c o s S - s i n / A s i n S cos(/A-8) =

tanx

t a n / \n 6 tan(>A + S ) =

sinx

tan(/\-B) = cotx =

cosx

1-tan>AtanB tan>A-tanB

Quotient Identities tanx =

c o s / \ c o s B +sin/A s I n B

1+ tan/\tan6

cosx sinx

Double-Angle Identities Pythagorean Identities sin^ x + cos^

X =

1

sin2x = 2 s i n x c o s x c o s 2 x = cos^ x - s i n P x = 2cos^x-1

1 + tan^ X = sec^ x 1 + cot^

X

= csc^

X

= 1-2sirfx 2 tanx tan 2 x = 1-tan^x

Odd and Even Identities sin(-x) =- sinx c s c ( - x ) =- c s c x

Half-Angle Identities

c o s (-x) = c o s x

sin— = ± 2

sec(-x) = s e c x tan(-x) =- t a n x

1 - cos

1 +cos COS — =

cot(-x) = - c o t x

2 tan — = 2

±

V

X

2

1-cosx sinx

X

sinx 1+ cosx

Cofunction Properties sin(f-x) = cosx cos(f-x) = sinx tan(f-x) = cotx cot(f-x) = tanx sec(f-x) = cscx csc(f-x) = secx

©2012, T E S C C C

09/11/12

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I n t h e f o l l o w i n g e x a m p l e s , we will u s e t r i g o n o m e t r i c i d e n t i t i e s t o s i m p l i f y expressions and t o prove o t h e r identities. Simplify each expression. 1)

tan^cos^

2)

tan(90° -

cos^cot(90°-^)

3)

(l-sinA-Xl + sinA-)

4)

5)

csc^ ^-(1 - cos^

6)

7)

s i n x cos A '

8)

(sinA' + cosA')^+

(sinA'-coSA')^

10)

cos^/ + c o s / s i n ^ /

1-cos^

9)

11)

cot^^ 1+ csc^

cos^ 1+sin^

j^^cgcg

1 + sin^ cos^