Compound Angle Identities

Question 1

What is the identity for Sin(A + B) ?

Answer 1

SinACosB + CosASinB

Question 2

What is the identity for Sin(A - B) ?

Answer 2

SinACosB - CosASinB

Question 3

What is the identity for Cos(A + B) ?

Answer 3

CosACosB - SinASinB

Question 4

What is the identity for Cos(A - B) ?

Answer 4

CosACosB + SinASinB

Question 5

What is the identity for Tan(A + B) ?

Answer 5

Sin(A + B) / Cos(A + B)

=SinACosB + CosASinB / CosACosB - SinASinB

=tanA + tanB / 1 - tanAtanB

Question 6

What is the identity for Tan(A - B) ?

Answer 6

Sin(A - B) / Cos(A -B)

= SinACosB - CosASinB / CosACosB + SinASinB

=tanA - tanB / 1 + tanAtanB

Question 7

How would you rewrite Sin30°Cos50° + Cos30°Sin50° as a single angle?

Answer 7

Sin(30° + 50°) = Sin(80°)

Question 8

Solve Sin x = Cos(x + 30°) for 0°≤ x ≤360 °

Answer 8

Sin x = CosxCos30°-SinxSin30°
Sin x + SinxSin30°=CosxCos30°
Sin x(1+Sin30)=CosxCos30°
Sin x / Cos x = Cos30° / 1 + Sin30°
Tan x = √3 / 3
tan-1(√3 / 3)
x=30°,210°

Question 9

Solve Cos3xCos2x - Sin3xSin2x = 0.5
for 0≤x≤π

Answer 9

Cos(3x+2x) = 0.5
Cos5x = 0.5
0≤x≤π becomes 0≤5x≤5π
Cos -1 (0.5) = 1/3π
2π -1/3π = 5/3π
Cos5x = 1/3π, 5/3π, 7/3π, 11/3π, 13/3π, 17/3π
divide by 5
Cos x = 1/15π, 1/3π, 7/15π, 11/15π, 13/15π

Question 10

What does Sin(A±B) equal?

Answer 10

Sin(A±B) = SinACosB ± CosASinB

Question 11

What does Tan(A±B) equal?

Answer 11

tanA ± tanB / 1 ∓ tanAtanB

Question 12

What does Cos(A±B) equal?

Answer 12

CosACosB ∓ SinASinB