Chapter 9.2, Problem 6AYU

$\cos \frac{2\pi }{3}=\_\_\_\_$. (pp. 385-387)

To determine

To fill: The value of $\text{cos}\frac{2\pi }{3}$.

Answer to Problem 6AYU

Solution:

$\text{cos}\frac{2\pi }{3} = - \frac{1}{2}$.

Given:

It is asked to fill up the value of $\text{cos}\frac{2\pi }{3}$.

Explanation of Solution

The exact values of the trigonometric functions of $\frac{2\pi }{3} = 60\text{°}$. Using symmetry, it can be easily find the exact values of the trigonometric functions of $\frac{\pi }{3} = 60\text{°}, \frac{2\pi }{3} = 120\text{°,} \frac{\text{4}\pi }{\text{3}} = 240\text{°}$ and $\frac{5\pi }{3} = 300\text{°}$. The point on the unit circle corresponding to $\frac{\pi }{3} = 60\text{°}$ is $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. Using symmetry with respect to the origin point $\left(- \frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ is the point on the unit circle that corresponds to the angle $\frac{2\pi }{3} = 120\text{°}$.

$\text{cos}\frac{2\pi }{3}$ corresponds to the $x$-coordinate of the point $\left(- \frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

$\text{cos}\frac{2\pi }{3} = - \frac{1}{2}$