Exercise 69 Page 520

Let's draw a unit circle showing $\cos \frac{\pi }{3}=\frac{1}{2}=0.5.$

If we reflect the right triangle in the $y\text{-}$axis, we can make a second statement. Notice that half a lap in the unit circle is $\pi$ rad. Therefore, by subtracting the reference angle from $\pi ,$ we get the measure of the reflected triangle's angle. We get the statement $\cos \frac{2\pi }{3}=\text{-}0.5.$ Next, we will reflect the new triangle in the $x\text{-}$axis. This triangle will create an angle on the unit circle that is $\frac{\pi }{3}$ greater than $\pi .$ We get the statement $\cos \frac{4\pi }{3}=\text{-}0.5.$ Finally, we will reflect the triangle in the third quadrant in the $y\text{-}$axis which gives us a third statement. This triangle will have an angle that is $\frac{\pi }{3}$ less than a full lap around the unit circle, $2\pi .$ We get the statement $\cos \frac{5\pi }{3}=0.5.$

Note that these are only a few examples of the values of cosine. We could as well subtract multiples of $\frac{\pi }{3}$ from $\frac{\pi }{3}$ instead of adding the multiples. Then, we would obtain values of cosine for different arguments.