Question

Find the values of the trigonometric functions:
i) $\sin {{765}^{\circ }}$
ii) $\csc (-1410)$

Answer 1

Hint: Here, we know that the trigonometric functions like sine and cosec functions are periodic functions. Therefore, we can write $\sin {{765}^{\circ }}$ as $\sin {{765}^{\circ }}=\sin (2\times {{360}^{\circ }}+{{45}^{\circ }})$ and similarly we can write $\csc (-1410)$ as, $\csc (-1410)=-\csc (4\times {{360}^{\circ }}-{{30}^{\circ }})$

Complete step by step solution:
Here, we have to find the value of $\sin {{765}^{\circ }}$ and $\csc (-1410)$.
i) $\sin {{765}^{\circ }}$
Now, first let us consider the function $\sin {{765}^{\circ }}$.
Here, first we should know about periodic functions.
We know that a periodic function is a function that repeats its value on regular intervals or periods. A function f is said to be periodic for a period t, if
$f(x+t)=f(x)$
We also know that the trigonometric functions sin x, cos x and tan x are periodic functions. The functions sin x and cos x have the period $2\pi $. Hence, we can say that,
$\begin{align}
  & \sin (2\pi +\theta )=\sin \theta \\
 & \cos (2\pi +\theta )=\cos \theta \\
\end{align}$
Here, we are given $\sin {{765}^{\circ }}$ and it can be written as:
$\sin {{765}^{\circ }}=\sin (2\times {{360}^{\circ }}+{{45}^{\circ }})$
Since, sine function is periodic, we can say that,
$\sin {{765}^{\circ }}=\sin {{45}^{\circ }}$
We know that the value of $\sin {{45}^{\circ }}=\dfrac{1}{\sqrt{2}}$.
Hence, we can say that, $\sin {{765}^{\circ }}=\dfrac{1}{\sqrt{2}}$

ii) $\csc (-1410)$
We also know that the cosec function, being a trigonometric function, is also a periodic function with period $2\pi $. The function will repeat after the intervals $2\pi ,4\pi ,6\pi ,..$
Here, we can write $\csc (-1410)$ as:
$\csc (-1410)=-\csc (1410)$
Since, we have $\csc (-x)=-\csc x$.
We know that,
$\begin{align}
  & \csc (8\pi -x)=-\csc x \\
 & \Rightarrow \csc (4\times 2\pi -x)=-\csc x \\
\end{align}$
Hence, we will get,
$\csc (-1410)=-\csc (4\times {{360}^{\circ }}-{{30}^{\circ }})$
Since, cosec function is a periodic function,
$\begin{align}
  & \csc (-1410)=-\csc (-{{30}^{\circ }}) \\
 & \Rightarrow \csc (-1410)=\csc ({{30}^{\circ }}) \\
\end{align}$
We have,
$\csc {{30}^{\circ }}=2$
Therefore, we will get the function as,
$\Rightarrow \csc (-1410)=2$
 Hence, we can say that the value of $\csc (-1410)=2$.

Note: Students generally get confused in the trigonometric formulae. Students get confused and write $\sin (2\pi +\theta )=\cos \theta $ and $\cos (2\pi +\theta )=\sin \theta $, which is wrong. This confusion should be avoided as it can lead to wrong answers. The formulae must be remembered properly.