Question

How do you find the six trigonometric functions of $540$ degree?

Answer 1

In this question we need to find the value of six trigonometric values of angle $540$ degree. To solve these questions we need to know the definition of trigonometric functions, period of trigonometric functions and knowing of common angle values of trigonometric functions to find the value at angle $540$ degree.

Complete step-by-step solution:
Let us try to solve the questions in which we are asked to find the value of trigonometric functions at angle $540$ degree. To solve this we need to know the period of trigonometric functions, period of trigonometric function sine, cosine, tangent, cosecant, secant and cotangent is $2\pi$ . Period of a trigonometric function is the value after trigonometric functions repeat its value. We know that $540$ degree in radians is equal to $3\pi$ .
${540^ \circ }$ = $3\pi$ $eq(1)$
As we know that period of sine function is $2\pi$ .So the value of $\sin ({540^ \circ }) = \sin (3\pi )$ by $eq(1)$ .
$\sin (3\pi ) = \sin (2\pi + \pi ) = \sin (\pi )$ ( $\because$ period of sine function is $2\pi$ )
And we know that value of $\sin (\pi ) = 0$ .Hence$\sin ({540^ \circ }) = 0$.
Similarly, the cosine function, as we know it, is $2\pi$ and $\cos (\pi ) = - 1$ .
So, $\cos ({540^ \circ }) = \cos (3\pi ) = \cos (2\pi + \pi ) = \cos (\pi ) = - 1$
Hence,$\cos ({540^ \circ }) = - 1$
Similarly, tangent function, as we know, period is $2\pi$ and $\tan (\pi ) = 0$ .
$\tan ({540^ \circ }) = \tan (3\pi ) = \tan (2\pi + \pi ) = \tan (\pi ) = 0$
Hence, $\tan ({540^ \circ }) = 0$
Now as we know that $\operatorname{cosec} \theta = \dfrac{1}{{\sin \theta }}$ , $\sec \theta = \dfrac{1}{{\cos \theta }}$ and $\cot \theta = \dfrac{1}{{\tan \theta }}$ .
Now, by using value of $\sin ({540^ \circ })$ , $\cos ({540^ \circ })$ , $\tan ({540^ \circ })$ and above properties, we get the value of other three trigonometric functions.
$\Rightarrow \operatorname{cosec} ({540^ \circ }) = \dfrac{1}{{\sin ({{540}^ \circ })}} = \dfrac{1}{0} = \infty$
$\Rightarrow \sec ({540^ \circ }) = \dfrac{1}{{\cos ({{540}^ \circ })}} = \dfrac{1}{{ - 1}} = - 1$
$\Rightarrow \cot ({540^ \circ }) = \dfrac{1}{{\tan ({{540}^ \circ })}} = \dfrac{1}{0} = \infty$
Hence we have found the values of all six trigonometric functions.

Note: We can also find the value$\operatorname{cosec} ({540^ \circ })$, $sec({540^ \circ })$and $\cot ({540^ \circ })$ by using their periodicity, similarly as we find the value of$\sin ({540^ \circ })$, $\cos ({540^ \circ })$and$\tan ({540^ \circ })$. Period of cosecant, secant and cotangent function is$2\pi$.