H Essential Trig

ESSENTIAL TRIGONOMETRIC IDENTITIES FOR PHYSICS & CALCULUS θ

For each real x, , , and α

β

that are elements of the domain of the specified functions, the following identities hold:

8. Complementary angle and 90° rotation formulas:

1. Definitions of derived trigonometric functions:

tan θ ≡

sin θ cos θ 1 1 1 ; cot θ ≡ ≡ ; sec θ ≡ ; csc θ ≡ cos θ sin θ tan θ cos θ sin θ

2. Even/odd properties of trigonometric functions:

cos ( π2 − θ ) ≡ sin θ

cos (θ ± π2 ) ≡ ∓ sin θ

sin ( π2 − θ ) ≡ cos θ

sin (θ ± π2 ) ≡ ± cos θ

tan ( π2 − θ ) ≡ cot θ

tan (θ ± π2 ) ≡ − cot θ

cos ( −θ ) ≡ cos (θ ) ;

sin ( −θ ) ≡ − sin (θ ) ;

tan ( −θ ) ≡ − tan (θ )

cot ( π2 − θ ) ≡ tan θ

cot (θ ± π2 ) ≡ − tan θ

cot ( −θ ) ≡ − cot (θ ) ;

sec ( −θ ) ≡ sec (θ ) ;

csc ( −θ ) ≡ − csc (θ )

sec ( π2 − θ ) ≡ csc θ

sec (θ ± π2 ) ≡ ∓ csc θ

csc ( π2 − θ ) ≡ sec θ

csc (θ ± π2 ) ≡ ± sec θ

3. Identities obtained from sin2 + cos2 = 1: θ

tan 2 θ + 1 ≡ sec 2 θ

θ

cot 2 θ + 1 ≡ csc2 θ

9. Supplementary angle and 180° rotation formulas: cos (π − θ ) ≡ − cos θ

cos (θ ± π ) ≡ − cos θ

cos (α ± β ) ≡ cos α cos β ∓ sin α sin β

sin (π − θ ) ≡ sin θ

sin (θ ± π ) ≡ − sin θ

sin (α ± β ) ≡ sin α cos β ± sin β cos α

tan (π − θ ) ≡ − tan θ

tan (θ ± π ) ≡ tan θ

cot ( π − θ ) ≡ − cot θ

cot (θ ± π ) ≡ cot θ

sec ( π − θ ) ≡ − sec θ

sec (θ ± π ) ≡ − sec θ

csc (π − θ ) ≡ csc θ

csc (θ ± π ) ≡ − csc θ

4. Sums or differences of angles:

tan (α ± β ) ≡

tan α ± tan β 1 ∓ tan α tan β

cot (α ± β ) ≡

cot α cot β ∓ 1 cot β ± cot α

sec (α ± β ) ≡

sec α sec β 1 ∓ tan α tan β

csc (α ± β ) ≡

csc α csc β cot β ± cot α

10. Squares of trigonometric functions:

5. Double angle formulas: cos ( 2θ ) ≡ cos 2 θ − sin 2 θ ≡ 2 cos 2 θ − 1 ≡ 1 − 2sin 2 θ sin ( 2θ ) ≡ 2sin θ cos θ tan ( 2θ ) ≡

2 tan θ 1 − tan 2 θ

cot ( 2θ ) ≡

cot 2 θ − 1 2 cot θ

sec ( 2θ ) ≡

sec 2 θ 1 − tan 2 θ

csc ( 2θ ) ≡

csc2 θ 2 cot θ

6. Sum-to-product/difference-to-product formulas: cos α + cos β ≡ 2 cos  12 (α + β )  cos  12 (α − β )  cos α − cos β ≡ 2 sin  12 ( β − α )  sin  12 ( β + α )  sin α ± sin β ≡ 2 sin  12 (α ± β )  cos  12 (α ∓ β )  tan α ± tan β ≡ tan (α ± β )(1 ∓ tan α tan β )

cos 2 θ ≡ 12 + 12 cos 2θ

sin 2 θ ≡ 12 − 12 cos 2θ

tan 2 θ ≡

1 − cos 2θ 1 + cos 2θ

cot 2 θ ≡

1 + cos 2θ 1 − cos 2θ

sec 2 θ ≡

2 1 + cos 2θ

csc 2 θ ≡

2 1 − cos 2θ

11. Half angle formulas:

cos tan

θ 2

θ 2

≡± ≡

1 2

(1 + cos θ )

sin

θ 2

1 2

≡±

(1 − cos θ )

sin θ 1 − cos θ 1 − cos θ ≡ ≡± 1 + cos θ sin θ 1 + cos θ

12. Compositions of trigonometric and inverse trigonometric functions (for −1 ≤ x ≤ 1 and 0 ≤ cos ( sin −1 x ) ≡ 1 − x 2

θ

≤ /2): π

1

cos ( tan −1 x ) ≡

1 + x2

7. Product-to-sum formulas: cos α cos β = 12 cos (α − β ) + cos (α + β )  sin α cos β = 12 sin (α + β ) + sin (α − β ) 

cos −1 ( sin θ ) ≡ sin ( tan −1 x ) ≡

sin α sin β = cos (α − β ) − cos (α + β )  1 2

tan ( cos −1 x ) ≡

VC DEPARTMENT OF MATHEMATICS

π 2

sin ( cos −1 x ) ≡ 1 − x 2

−θ x

1+ x

2

1 − x2 x

sin −1 ( cos θ ) ≡ tan ( sin −1 x ) ≡

π 2

−θ x

1 − x2

REVISED SUMMER 2007