Question

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

2 cos² 5θ = 1

Answer 1

Given:

2 cos² 5θ = 1

Step 1: Simplify the equation

2 cos² 5θ = 1   ...[Divide both sides by 2]

`cos^2 (50) = 1/2`

Step 2: Take the square root of both sides.

cos 50 = ± `sqrt(1/2) = ± 1/sqrt2`

Step 3:

Since θ is acute, 0° < θ < 90° ⇒ 0° < 5θ < 450° 

Step 4: Find the angle whose cosine is `1/sqrt2`.   ...[taking the principal (smallest positive) angle where cos⁡ x]

cos 45° = `1/sqrt2`.

Step 5: Equate the angles and solve for θ.

5θ = 45°

θ = `(45°)/5`

θ = 9°

Therefore, the measure of angle θ is 9°.