Question

Use even or odd properties of the trigonometric function to find the exact value of the expression. $tan ( - \frac{\pi}{6} )$

Answer 1

We can rewrite the expression as following:

$\tan (-\frac{\pi}{6})=\frac{\sin(-\frac{\pi}{6})}{\cos(-\frac{\pi}{6})}$

Since sine is odd, and cosine is even,

$\frac{\sin(-\frac{\pi}{6})}{\cos(-\frac{\pi}{6})}\rightarrow\frac{-\sin(\frac{\pi}{6})}{\cos(\frac{\pi}{6})}\rightarrow-\frac{\sin(\frac{\pi}{6})}{\cos(\frac{\pi}{6})}$

Evaluating both trigonometric functions,

$-\frac{\sin(\frac{\pi}{6})}{\cos(\frac{\pi}{6})}=-\frac{\frac{1}{2}}{\frac{\sqrt[]{3}}{2}}\rightarrow-\frac{2}{2\sqrt[\square]{3}}\rightarrow-\frac{2\sqrt[\square]{3}}{6}\rightarrow-\frac{\sqrt[]{3}}{3}$

Thereby,

$\tan (-\frac{\pi}{6})=-\frac{\sqrt[]{3}}{3}$