Question

find all solutions to the equation.

\\( \cos \theta = \frac { \sqrt { 3 } } { 2 } \\)

write your answer in radians in terms of \\( \pi \\), and use the \or\ button as necessary.

example: \\( \theta = \frac { \pi } { 5 } + 2 k \pi, k \in \mathbb { z } \\) or \\( \theta = \frac { \pi } { 7 } + k \pi, k \in \mathbb { z } \\)

\\( \theta = \\)

Explanation:

Step1: Recall the cosine values

We know that \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\).

Step2: Use the periodicity of cosine function

The cosine function \(y = \cos\theta\) has a period of \(2\pi\). Also, \(\cos\theta=\cos(2k\pi\pm\alpha)\) for \(k\in\mathbb{Z}\). Since \(\cos\theta=\frac{\sqrt{3}}{2}\) and \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\), the general solutions are \(\theta = 2k\pi\pm\frac{\pi}{6},k\in\mathbb{Z}\).

Answer:

\(\theta=\frac{\pi}{6}+2k\pi\) or \(\theta =-\frac{\pi}{6}+2k\pi,k\in\mathbb{Z}\)