Question

Find the value of $\cot \left( { - \dfrac{{15\pi }}{4}} \right)$

Answer 1

Hint: Note that, \[{\text{cot}}( - x) = - \cot x\]
Again, we know that the function y= cotx has a period of\[\pi \] or \[180^\circ .\], i.e. the value of cotx repeats after an interval of \[\pi \]or 180°.
Therefore write $\dfrac{{15\pi }}{4}$ as \[4\pi - \dfrac{\pi }{4}\] and proceed.

Complete step-by-step answer:
We know that the function y= cotx has a period of \[\pi \] or \[180^\circ .\], i.e. the value of cotx repeats after an interval of \[\pi \] or \[180^\circ .\].
Therefore,
\[\cot \left( { - \dfrac{{15\pi }}{4}} \right)\]
Since, \[{\text{cot}}( - x) = - \cot x\],
\[ = - \cot \left( {\dfrac{{15\pi }}{4}} \right)\]
On expanding the numerator we get,
\[ = - \cot \left( {\dfrac{{16\pi - \pi }}{4}} \right)\]
On simplification we get,
\[ = - \cot \left( {4\pi - \dfrac{\pi }{4}} \right)\]
Since, \[\left( {\dfrac{{15\pi }}{4}} \right){\text{ }}\] lies in the fourth quadrant, therefore \[\cot \left( {\dfrac{{15\pi }}{4}} \right){\text{ }}\] will be negative
\[ = - \left( { - \cot \dfrac{\pi }{3}} \right)\]
\[ = \cot \dfrac{\pi }{3}\]
As, \[\cot \dfrac{\pi }{3} = \dfrac{1}{{\sqrt 3 }}\]
\[ = \dfrac{1}{{\sqrt 3 }}\]
Therefore the value of $\cot \left( { - \dfrac{{15\pi }}{4}} \right)$is \[\dfrac{1}{{\sqrt 3 }}\].

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