Question

Find the values of the trigonometric function: $cosec\left( -{{1410}^{\circ }} \right)$.

Answer 1

Hint: In this question, we need to use the fact the trigonometric ratios remain unchanged if the angle is added by a multiple of ${{360}^{\circ }}$ or $2\pi$. Therefore, we should try to add or subtract a multiple of ${{360}^{\circ }}$ to the given angle to obtain an angle between ${{0}^{\circ }}$ and ${{360}^{\circ }}$. Then, using the trigonometric ratios of standard angles, we can obtain the answer to the given question.

Complete step-by-step answer:
We know that the angle between lines can range can be expressed between ${{0}^{\circ }}$ to ${{360}^{\circ }}$ because when we increase the angle between two lines, we can to more than assume one of the lines to be along the x-axis and the other line to revolve with respect to the origin. Therefore, as ${{360}^{\circ }}$ represents a full rotation, when the angle increases above ${{360}^{\circ }}$, then the revolving line returns to the earlier positions described by the angle between ${{0}^{\circ }}$ and ${{360}^{\circ }}$.
Therefore, as the trigonometric ratios depend on the angles, the trigonometric ratios are unchanged if the angle is increased or decreased by a multiple of ${{360}^{\circ }}$. Therefore, as cosec is also a trigonometric ratio, mathematically we can write
$\text{cosec}\left( n\times {{360}^{\circ }}+\theta \right)=\text{cosec}\left( \theta \right)\text{ where n}\in Z.................(1.1)$
where n is an integer.
Therefore, in this question, using equation (1.1), we can write
$\text{cosec}\left( -{{1410}^{\circ }} \right)=\text{cosec}\left( -4\times {{360}^{\circ }}+{{30}^{\circ }} \right)=\text{cosec}\left( {{30}^{\circ }} \right)..........(1.2)$
Now, we know that the definition of cosec is given by
$\text{cosec}\left( \theta \right)=\dfrac{1}{\sin \left( \theta \right)}.................(1.3)$
Also, we know that the sine of ${{30}^{\circ }}$ will be given by
$\sin \left( {{30}^{\circ }} \right)=\dfrac{1}{2}$
Using this value in equation (1.2) and (1.3), we get
$\text{cosec}\left( {{30}^{\circ }} \right)=\dfrac{1}{\sin \left( {{30}^{\circ }} \right)}=\dfrac{1}{\dfrac{1}{2}}=2$
Thus, the answer to the given question should be equal to 2.

Note: In equation (1.2), we could also have written that $\text{cosec}\left( -{{1410}^{\circ }} \right)=\text{cosec}\left( -3\times {{360}^{\circ }}-{{330}^{\circ }} \right)=\text{cosec}\left( -{{330}^{\circ }} \right)$
However, as $\sin \left( -{{330}^{\circ }} \right)=\dfrac{1}{2}$, using equation (1.3) we would get the same answer as obtained in the solution above.