Question

Find the value exact value of tan 105 degrees. Use a half-angle identity.

Answer 1

Answer:

The exact value of $tan(105\°)$ is $-2-\sqrt{3}$

Step-by-step explanation:

The half-angle identity for 'tangent' is.....

$tan(\frac{\theta}{2})=\frac{sin(\theta)}{1+cos(\theta)}$

Here, $\frac{\theta}{2}=105\° \Rightarrow \theta=210\°$

Plugging the value of $\theta$ into the above formula.....

$tan(\frac{210\°}{2})=\frac{sin(210\°)}{1+cos(210\°)}\\ \\ tan(105\°)=\frac{-\frac{1}{2}}{1+(-\frac{\sqrt{3}}{2})}\\ \\ tan(105\°)=\frac{-\frac{1}{2}}{1-\frac{\sqrt{3}}{2}}\\ \\ tan(105\°)=\frac{-\frac{1}{2}}{\frac{2-\sqrt{3}}{2}}\\ \\tan(105\°)=-\frac{1}{2-\sqrt{3}}=-\frac{(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}=-\frac{(2+\sqrt{3})}{4-3}=-\frac{(2+\sqrt{3})}{1}\\ \\ tan(105\°)=-2-\sqrt{3}$

Answer 2

The simplified answer after all the work is -2-the square root of 3...