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

Question

Use even or odd properties of the trigonometric function to find the exact value of the expression.
`tan ( - (\pi)/(6) )`

Answer 1

We can rewrite the expression as following:

`\tan (-(\pi)/(6))=(\sin(-(\pi)/(6)))/(\cos(-(\pi)/(6)))`

Since sine is odd, and cosine is even,

`(\sin(-(\pi)/(6)))/(\cos(-(\pi)/(6)))\rightarrow(-\sin((\pi)/(6)))/(\cos((\pi)/(6)))\rightarrow-(\sin((\pi)/(6)))/(\cos((\pi)/(6)))`

Evaluating both trigonometric functions,

`-(\sin((\pi)/(6)))/(\cos((\pi)/(6)))=-\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 (-(\pi)/(6))=-\frac{\sqrt[]{3}}{3}`