TRIGONOMETRIC IDENTITIES: USEFUL FORMULAS

TRIGONOMETRIC IDENTITIES: USEFUL FORMULAS Tangent Identities

Double Angle Identities

tan 𝜃 =

sin 𝜃 cos 𝜃

cot 𝜃 =

cos 𝜃 sin 𝜃

sin(2𝜃) = 2 sin 𝜃 cos 𝜃 cos(2𝜃) = cos2 𝜃 − sin2 𝜃

Reciprocal Identities cos(2𝜃) = 2cos2 𝜃 − 1 1 csc 𝜃 = sin 𝜃

1 sin 𝜃 = csc 𝜃

cos(2𝜃) = 1 − 2 sin2 𝜃

sec 𝜃 =

1 cos 𝜃

cos 𝜃 =

1 sec 𝜃

cot 𝜃 =

1 tan 𝜃

tan 𝜃 =

1 cot 𝜃

tan(2𝜃) =

Power-Reducing and Half-Angle Identities

Pythagorean Identities sin2 𝜃 + cos2 𝜃 = 1

1 + tan2 𝜃 = sec 2 𝜃

sin2 𝜃 = 1 − cos2 𝜃

tan2 𝜃 = sec 2 𝜃 − 1

cos2 𝜃 = 1 − sin2 𝜃

1 + cot 2 𝜃 = csc 2 𝜃

cot 2 𝜃 = csc 2 𝜃 − 1

sin2 𝜃 =

1 − cos 2𝜃 2

1 − cos 2𝜃 sin 𝜃 = ±√ 2

cos2 𝜃 =

1 + cos 2𝜃 2

1 + cos 2𝜃 cos 𝜃 = ±√ 2

tan2 𝜃 =

Even Identities cos(−𝜃) = cos 𝜃

sec(−𝜃) = sec 𝜃 tan 𝜃 = ±√

Odd Identities sin(−𝜃) = − sin 𝜃

csc(−𝜃) = − csc 𝜃

tan(−𝜃) = − tan 𝜃

cot(−𝜃) = − cot 𝜃

𝜋 sin ( − 𝜃) = cos 𝜃 2

𝜋 cos ( − 𝜃) = sin 𝜃 2

𝜋 tan ( − 𝜃) = cot 𝜃 2

𝜋 cot ( − 𝜃) = tan 𝜃 2

𝜋 sec ( − 𝜃) = csc 𝜃 2

𝜋 csc ( − 𝜃) = sec 𝜃 2

Product-to-Sum Identities 1 sin 𝐴 cos 𝐵 = [sin(𝐴 + 𝐵) + sin(𝐴 − 𝐵)] 2 sin 𝐴 sin 𝐵 =

1 [cos(𝐴 − 𝐵) − cos(𝐴 + 𝐵)] 2

1 cos 𝐴 sin 𝐵 = [sin(𝐴 + 𝐵) − sin(𝐴 − 𝐵)] 2 cos 𝐴 cos 𝐵 =

1 [cos(𝐴 − 𝐵) + cos(𝐴 + 𝐵)] 2

Sum-to-Product Identities

Sum and Difference Identities sin(𝐴 + 𝐵) = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵 sin(𝐴 − 𝐵) = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵 cos(𝐴 + 𝐵) = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵 cos(𝐴 − 𝐵) = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵 tan 𝐴 + tan 𝐵 1 − tan 𝐴 tan 𝐵

1 − cos 2𝜃 1 + cos 2𝜃

1 − cos 2𝜃 1 − cos 2𝜃 sin 2𝜃 = = 1 + cos 2𝜃 sin 2𝜃 1 + cos 2𝜃

Cofunction Identities

tan(𝐴 + 𝐵) =

2 tan 𝜃 1 − tan2 𝜃

tan(𝐴 − 𝐵) =

tan 𝐴 − tan 𝐵 1 + tan 𝐴 tan 𝐵

𝐴+𝐵 𝐴−𝐵 sin 𝐴 + sin 𝐵 = 2 sin ( ) cos ( ) 2 2 𝐴−𝐵 𝐴+𝐵 sin 𝐴 − sin 𝐵 = 2 sin ( ) cos ( ) 2 2 𝐴+𝐵 𝐴−𝐵 cos 𝐴 + cos 𝐵 = 2 cos ( ) cos ( ) 2 2 𝐴+𝐵 𝐴−𝐵 cos 𝐴 − cos 𝐵 = −2 sin ( ) sin ( ) 2 2

TRIGONOMETRIC IDENTITIES: USEFUL FORMULAS

THE UNIT CIRCLE 𝑥2 + 𝑦2 = 1 (𝑥, 𝑦) = (cos 𝜃 , sin 𝜃)