Question

Find the value of $\operatorname{cosec} ( - {1410^ \circ })$

Answer 1

Hint: We know that \[{\text{cosec}}\left( { - \theta } \right) = - {\text{cosec}}\theta \]
Again, the function \[y = \cos ecx\] has a period of \[2\pi \] or \[360^\circ \], i.e. the value of \[\cos ecx\;\] repeats after an interval of \[2\pi \] or \[360^\circ \].
Therefore write \[1410^\circ \] as \[(4 \times 2\pi - 30^\circ )\] and proceed.

Complete step-by-step answer:
We know that the function \[y = \cos ecx\] has a period of \[2\pi \] or \[360^\circ \], i.e. the value of \[\cos ecx\;\] repeats after an interval of \[2\pi \] or \[360^\circ \].

Therefore,
\[\operatorname{cosec} ( - {1410^ \circ })\]
Using, \[\left[ {{\text{cosec}}\left( { - \theta } \right) = - {\text{cosec}}\theta } \right]\], we get,
\[ = - \operatorname{cosec} ({1410^ \circ }){\text{ }}\]
We can write the above statement as,
\[ = - \operatorname{cosec} \left( {(4 \times {{360}^ \circ }) - {{30}^ \circ }} \right)\]
Since \[{\text{141}}{0^ \circ }\] lies in the fourth quadrant, therefore is \[{\text{cosec141}}{0^ \circ }\] negative
\[ = - \left( { - \operatorname{cosec} {{30}^ \circ }} \right){\text{ }}\]
\[ = \operatorname{cosec} {30^ \circ }\]
As, \[\operatorname{cosec} ({30^ \circ }) = 2\], we get,
\[ = 2\]
Hence, the value of $\operatorname{cosec} ( - {1410^ \circ })$ is 2.

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
\[\cos x\]	1	$\dfrac{{\sqrt 3 }}{2}$	$\dfrac{1}{{\sqrt 2 }}$	$\dfrac{1}{2}$	0
\[\tan x\]	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