### Home > INT3 > Chapter 11 > Lesson 11.2.4 > Problem11-104

11-104.

Solve each equation for $0\le x<2\pi$. No calculator should be necessary.

1. $\cos(x)=-\frac{1}{2}$

$\text{Recall that } \cos \left( \frac{\pi}{6}\right) = \frac{1}{2},$
$\text{ so the reference angle will be } 30^\circ.$

In which quadrants is cosine negative? Sketch a unit circle with right triangles with $30^\circ$ angles at the origin in the second and third quadrants. What are the circular angles?

$x = 120^\circ$ and $240^\circ$

Now convert the angles into radians.

$120\left(\frac{\pi}{180}\right)\ \ \ \ \ \ \ \ 240\left(\frac{\pi}{180}\right)$

$x = \frac{2{\pi}}{3}, \frac{4{\pi}}{3}$

1. $\tan(x)=\frac{\sqrt{3}}{3}$

See part (a).

1. $\operatorname { sin } ( x ) = 0$

If $\text{sin}\left(x\right) = 0, x = 0$.

$x = 0$

1. $\operatorname { cos } ( x ) = \frac { \sqrt { 2 } } { 2 }$

Remember the side ratios of $45^\circ$ right triangles.