Question

Find θ if 0° < θ ≤ 90°.

cos θ = 4 cos³ 30° − 3 cos 30°

Answer 1

Given:

0° < θ ≤ 90°

cos θ = 4 cos³ 30° − 3 cos 30°

Step 1: Triple angle identity

cos° = `sqrt3/2`

cos θ = `4(sqrt3/2)^3 − 3 (sqrt3/2)`

Step 2: Simplify

`(sqrt3/2)^3 = (sqrt3/2^3)^3 = (3 sqrt3)/8`

cos θ = `4((3 sqrt3)/8) − (3sqrt3)/2`

Step 3:

cos θ = `(12sqrt3)/8 − (3sqrt3)/2`

Step 4:

Common denominator = 8

`(3sqrt3)/2 = (12sqrt3)/8`

cos θ = `12sqrt3/8 − (12sqrt3)/8 = 0`

cosθ = 0 ⇒ θ = 90°   ...[0° < θ ≤ 90°]

θ = 90°