Question

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

4 cos² θ − 3 = 0

show that 4 cos³ θ − 3 cos θ = cos 3θ

Answer 1

Given:

`4 cos^2 θ − 3 = 0 → cos^2 θ = 3/4`

cos θ = ± `sqrt3/2`

since θ is acute (0° < θ < 90°),  cos θ > 0

cos⁡ θ = `sqrt3/2`

θ = 30°

Using angle formulas:

cos 3θ​ = cos(2θ + θ)

= cos 2θ cos θ − sin 2θ sin θ

= (2 cos² θ − 1) cos θ − (2 sin θ cos θ) sin θ

= 2 cos³ θ − cos θ − 2(1 − cos² θ) cos θ

= 4 cos³ θ − 3 cos θ

with θ = 30° : 4 cos³ θ − 3 cos θ = 0 = cos 90° = cos 3θ, consistent.