Calculus I - Trig Functions

Section 1.3 : Trig Functions

4. Determine the exact value of $\displaystyle \cos \left( { - \frac{{2\pi }}{3}} \right)$ without using a calculator.

Show All Steps Hide All Steps

Start Solution

First we can notice that $ - \pi + \frac{\pi }{3} = - \frac{{2\pi }}{3}$ so (recalling that negative angles rotate clockwise and positive angles rotation counter clockwise) the terminal line for $ - \frac{{2\pi }}{3}$ will form an angle of $\frac{\pi }{3}$ with the negative $x$-axis in the third quadrant and we’ll have the following unit circle for this problem.

$A unit circle with the angles $\frac{\pi }{6}$, $\frac{\pi }{4}$ and $\frac{\pi }{3}$ drawn in the first quadrant and labeled with the coordinates $\left( {\frac{{\sqrt 3 }}{2},\frac{1}{2}} \right)$, $\left( {\frac{{\sqrt 2 }}{2},\frac{{\sqrt 2 }}{2}} \right)$ and $\left( {\frac{1}{2},\frac{{\sqrt 3 }}{2}} \right)$ where they meet the circle. The axes are labeled with the angles 0 and $2\pi $, $\frac{\pi }{2}$, $\pi $ and $\frac{{3\pi }}{2}$ and the points (1,0), (0,1), (-1,0) and (0,-1). The angle $ - \frac{{2\pi }}{3}$ is drawn in the third quadrant and lies on the same line through the origin as the angle $\frac{\pi }{3}$, so the coordinates of the two points on the circle differ only in sign.$

Show Step 2

The line representing $ - \frac{{2\pi }}{3}$ is a mirror image of the line representing $\frac{\pi }{3}$ and so the coordinates for $ - \frac{{2\pi }}{3}$ will be the same as the coordinates for $\frac{\pi }{3}$ except that both coordinates will now be negative. So, our new coordinates will then be $\left( { - \frac{1}{2}, - \frac{{\sqrt 3 }}{2}} \right)$ and so the answer is,

\[\cos \left( { - \frac{{2\pi }}{3}} \right) = - \frac{1}{2}\]