Question

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

Answer 1

Hint: 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