How do you calculate the absolute value of trigonometric functions?

Question

How do you calculate:

$|\tan t|= \sqrt{3}$

The answer to this is required to be in the interval $[0,2\pi]$

Answer 1

$$|\tan(t)|=\sqrt{3} \ \Longrightarrow \ \tan(t)=\pm \sqrt{3}.$$ To solve this, you can look up the values, because they are ones which you will eventually come to remember. We know that if $\tan(t)=\sqrt{3}$ then $$t=\arctan(\sqrt{3})=\frac{\pi}{3}$$and, similarly, $$t=\arctan(-\sqrt{3})=-\frac{\pi}{3}=\frac{5\pi}{3}.$$