Question

Find the value of $\tan \dfrac{{19\pi }}{3}$

Answer 1

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 write $\dfrac{{19\pi }}{3}$ as $\left( {6\pi + \dfrac{\pi }{3}} \right)$ and proceed.

Complete step-by-step answer:
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,
$tan\dfrac{{19\pi }}{3}$
Above expression can be written as,
$= tan\left( {\dfrac{{18\pi + \pi }}{3}} \right)$
On separating the terms we get,
$= tan\left( {6\pi + \dfrac{\pi }{3}} \right)$
Since, $\dfrac{{{\text{19}}\pi }}{{\text{3}}}$lies in the first quadrant, therefore ${\text{tan}}\dfrac{{{\text{19}}\pi }}{{\text{3}}}$ will be positive,
$tan\dfrac{\pi }{3} = \sqrt 3$
As $tan\dfrac{\pi }{3} = \sqrt 3$,
$= \sqrt 3$
Therefore the value of $\tan \dfrac{{19\pi }}{3}$is $\sqrt 3$

Note: Note the following important formulae:
1.$\cos x = \dfrac{1}{{\sec x}}$ , $\sin x = \dfrac{1}{{\cos ecx}}$ , $\tan x = \dfrac{1}{{\cot x}}$
2.${\sin ^2}x + {\cos ^2}x = 1$
3.${\sec ^2}x - {\tan ^2}x = 1$
4.${\operatorname{cosec} ^2}x - {\cot ^2}x = 1$
5.$\sin ( - x) = - \sin x$
6.$\cos ( - x) = \cos x$
7.$\tan ( - x) = - \tan x$
8.$\sin \left( {2n\pi \pm x} \right) = \sin x{\text{ , period 2}}\pi {\text{ or 3}}{60^ \circ }$
9.$\cos \left( {2n\pi \pm x} \right) = \cos x{\text{ , period 2}}\pi {\text{ or 3}}{60^ \circ }$
10.$\tan \left( {n\pi \pm x} \right) = \tan x{\text{ , period }}\pi {\text{ or 18}}{0^ \circ }$
Sign convention:

Also, the trigonometric ratios of the standard angles are given by

| $0^\circ$| $30^\circ$| $45^\circ$| $60^\circ$| $90^\circ$
---|---|---|---|---|---
$\operatorname{Sin} x$| 0| $\dfrac{1}{2}$ | $\dfrac{1}{{\sqrt 2 }}$ | $\dfrac{{\sqrt 3 }}{2}$ | 1
$\operatorname{Cos} x$| 1| $\dfrac{{\sqrt 3 }}{2}$| $\dfrac{1}{{\sqrt 2 }}$| $\dfrac{1}{2}$| 0
$\operatorname{Tan} x$| 0| $\dfrac{1}{{\sqrt 3 }}$ | 1| $\sqrt 3$| Undefined
$Cotx$| undefined| $\sqrt 3$| 1| $\dfrac{1}{{\sqrt 3 }}$| 0
$\cos ecx$| undefined| 2| $\sqrt 2$| $\dfrac{2}{{\sqrt 3 }}$| 1
$\operatorname{Sec} x$| 1| $\dfrac{2}{{\sqrt 3 }}$| $\sqrt 2$| 2| Undefined