Question

find all solutions to the equation.
cosθ = - 1
write your answer in radians in terms of π, and use the or button as necessary.
example: θ = π/5 + 2kπ, k ∈ z or θ = π/7 + kπ, k ∈ z
θ =

Explanation:

Step1: Recall the unit - circle values

We know that on the unit circle, \(\cos\theta=-1\) when \(\theta = \pi\) (since \(\cos\pi=-1\)).

Step2: Consider the period of the cosine function

The period of the cosine function \(y = \cos\theta\) is \(2\pi\). So if \(\cos\theta=-1\) for \(\theta=\pi\), then for any integer \(k\), \(\cos(\pi + 2k\pi)=\cos\pi\cos(2k\pi)-\sin\pi\sin(2k\pi)\). Since \(\cos(2k\pi) = 1\) and \(\sin(2k\pi)=0\) and \(\cos\pi=-1,\sin\pi = 0\), we have \(\cos(\pi+2k\pi)=-1\).

Answer:

\(\theta=\pi + 2k\pi,k\in\mathbb{Z}\)