Question

An equation is given. (Enter your answers as a comma-separated list. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.) cos θ 2 − 1 = 0 (a) Find all solutions of the equation.

Answer 1

Answer:

$\theta = n\pi$

where $n = ..., -2,-1,0,1,2...$

Step-by-step explanation:

given equation is:

$\cos{2\theta} - 1 = 0$

since no range is provided we can solve for all values of $\theta$:

$\cos{2\theta} = 1$

$2\theta = \cos^{-1}{(1)}$

$2\theta = 0, 2\pi$ for one cycle of cos $(0 \leq \theta \leq 2\pi)$

$2\theta = 0, 2\pi, 4\pi, 6\pi ... 2n\pi$ for all cycles of cos

we should also include negative values.

$2\theta = -4\pi,-2\pi,0, 2\pi, 4\pi,... 2n\pi$

we can divide each value by 2, to get the solutions for $\theta$ instead of $2\theta$

Answer:

$\theta = -2\pi,-\pi,0, \pi, 2\pi,... n\pi$

This is the solution of the equation $\cos{2\theta} - 1 = 0$.

In its most general form we can write all solutions of the equation in terms of $n$

$\theta = n\pi$ where $n = ..., -2,-1,0,1,2...$ or n is an integer.