Question

Find the measure of angle θ in the following when θ is acute.

2 sin θ − 1 = 0

show that sin 3θ = 3 sin θ − 4 sin³ θ

Answer 1

Given:

2 sin θ − 1 = 0 ⇒ sin θ = `1/2`

Acute ⇒ θ ∈ (0∘,90∘), so

θ = 30°

Use sum/double-angle formulas.

sin 3θ​ = sin (2θ + θ)

= sin 2θ cos θ + cos 2θ sin θ

= (2 sin θ cos θ) cos θ + (1 − 2 sin² θ) sin θ

= 2 sin θ cos² θ + sin θ − 2 sin³ θ

= 2 sin θ(1 − sin² θ) + sin θ − 2 sin³θ

= 2 sin θ − 2 sin³ θ + sin θ − 2 sin³ θ

= 3 sinθ − 4 sin³ θ