Question

Find all possible values of $\cos(\theta)$ if

$$ \cos(2\theta) = 2\cos(\theta) .$$

a) $\cos(\theta) = 1$
b) $\cos(\theta) = -1$
c) $\cos(\theta) = \frac{1}{2}$
d) $\cos(\theta) = -\frac{1}{2}$

Answer 1

To solve the equation cos(2θ) = 2cos(θ), we use the double angle formula and transform it into a quadratic equation in terms of cos(θ). The solutions are cos(θ) = 1 and cos(θ) = -1/2, hence options a) and d) are correct.

Step-by-step explanation:

The student is asked to find all possible values of cos(θ) given that cos(2θ) = 2cos(θ). By applying the double angle formula for cosine, we know that cos(2θ) = cos^2(θ) - sin^2(θ) and it can also be written as 2cos^2(θ) - 1 by using the Pythagorean identity sin^2(θ) + cos^2(θ) = 1. Therefore, the provided equation transforms into 2cos^2(θ) - 1 = 2cos(θ), or 2cos^2(θ) - 2cos(θ) - 1 = 0. Solving this quadratic equation in terms of cos(θ), we get the potential solutions.

Solving this leads to two possible values for cos(θ): cos(θ) = 1 and cos(θ) = -½. These solutions correspond to the angles where the original equation is satisfied. The angle θ equals 0° or 360° for cos(θ) = 1, and θ equals 120° or 240° for cos(θ) = -½. Therefore, options a) and d) are the correct values for cos(θ).