Question

In the interval 0° ≤ θ < 360°, the solution set to the equation cos 2θ = cos θ includes all of the following angles except:

a) 45°
b) 135°
c) 225°
d) 315°

Answer 1

The solution set to the equation cos 2θ = cos θ includes all angles except 45°, 135°, 225°, and 315°.

Step-by-step explanation:

The solution set to the equation cos 2θ = cos θ can be found by first using the double angle formula for cosine: cos 2θ = 2cos^2(θ) - 1. Substituting this into the equation, we get 2cos^2(θ) - 1 = cos θ.

Now, we can simplify this equation:

2cos^2(θ) - cos θ - 1 = 0

Factoring this equation, we have (2cos θ + 1)(cos θ - 1) = 0. Therefore, the cos θ = -1/2 or cos θ = 1.

Now, we can find the angles for which these equations are true:

For cos θ = -1/2, the angle θ can be found using the inverse cosine function: θ = 120° or 240°.

For cos θ = 1, the angle θ can be found using the inverse cosine function: θ = 0°.

Therefore, the solution set to the equation cos 2θ = cos θ includes all angles except 45°, 135°, 225°, and 315°.