SOLVING TRIG EQUATIONS WITH COMPOUND ANGLES

Solve each equation for θ in the interval 0 ≤ θ ≤ 360.

Give your answers to 1 decimal place where appropriate.

Problem 1 :

sin θ° cos 15° + cos θ° sin 15° = 0.4

Solution:

sin θ° cos 15° + cos θ° sin 15° = 0.4

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

sin (θ + 15) = 0.4

θ + 15 = sin^-1(0.4)

θ + 15 = 23.57°

θ = 23.57° - 15°

θ = 8.57°

Problem 2 :

Solution:

Problem 3 :

cos (𝜃 - 60)° = sin 𝜃°

Solution:

cos (𝜃 - 60)° = sin 𝜃°

cos (𝜃 - 60)° = cos (90° - 𝜃)

(𝜃 - 60)° = 90° - 𝜃

𝜃 + 𝜃 = 90 + 60

2𝜃 = 150

𝜃 = 150/2

𝜃 = 75°

Problem 4 :

sin (𝜃 + 30)° = cos (𝜃 - 45)° 

Solution:

sin (𝜃 + 30)° = cos (𝜃 - 45)° 

cos (90 - (𝜃 + 30)) = cos (𝜃 - 45)° 

cos (90 - 𝜃 - 30) = cos (𝜃 - 45)° 

cos(60 - 𝜃) = cos (𝜃 - 45)

60 - 𝜃 = 𝜃 - 45

2𝜃 = 105

𝜃 = 105/2

𝜃 = 52.5°

Problem 5 :

3 cos (2𝜃 + 60)° - sin (2𝜃 - 30)° = 0

Solution: