Question

Find the value of $$tan{\displaystyle \frac{19\pi }{3}}$$.

Answer 1

Find the value of given trigonometric ratio

Given trigonometric ratio: $$tan{\displaystyle \frac{19\pi }{3}}$$

We know that the function $$y=\tan x$$ has a period of $\pi$ or $${180}^{\circ }$$, i.e. the value of $$\tan x$$ repeats after an interval of $\pi$ or$${180}^{\circ }$$.

Therefore, the given trigonometric ratio can be written as,

$$\begin{gathered} tan\frac{19\pi }{3}=tan\left(\frac{18\pi +\pi }{3}\right) \\ =tan\left(\frac{18\pi }{3}+\frac{\pi }{3}\right) \\ =tan\left(6\pi +\frac{\pi }{3}\right) \\ =tan\frac{\pi }{3} \\ =tan{60}^{\circ } \\ =\sqrt{3} \end{gathered}$$
Therefore, the value of $\tan {\displaystyle \frac{19\pi }{3}}$is $$\sqrt{3}$$.