Question

find all solutions to ( 2 cos ( \theta ) = - sqrt { 2 } ) on the interval ( 0 leq \theta < 2 pi ).

( \theta = )

give your answers as exact values in a list separated by commas.

question help: video 1 video 2

Explanation:

Step1: Solve for \(\cos(\theta)\)

Divide both sides of \(2\cos(\theta)=-\sqrt{2}\) by \(2\).
\(\cos(\theta)=\frac{-\sqrt{2}}{2}\)

Step2: Find \(\theta\) values

We know that \(\cos(\theta)=\frac{-\sqrt{2}}{2}\). The reference - angle \(\alpha\) for which \(\cos(\alpha)=\frac{\sqrt{2}}{2}\) is \(\alpha = \frac{\pi}{4}\).
Since \(\cos(\theta)<0\), \(\theta\) is in the second or third quadrant.
In the second quadrant, \(\theta=\pi-\frac{\pi}{4}=\frac{3\pi}{4}\).
In the third quadrant, \(\theta=\pi+\frac{\pi}{4}=\frac{5\pi}{4}\).

Answer:

\(\frac{3\pi}{4},\frac{5\pi}{4}\)