Answer

Verified

416.4k+ views

**Hint:**Here in the question we are supposed to find the inverse trigonometric function to find the angle. This means the value in the bracket of trigonometric functions given here is an exact value of the function at a certain angle. So we have to find that angle in this question. For that we will see the range for Cosecant and then check for the principal values of trigonometric functions.

**Complete step by step solution:**

Let us suppose that $y = {\csc ^{ - 1}}( - 1)$ ----equation (1)

As we can see we have converted this into an equation by putting two expressions on both sides of the sign ‘equals to’ (=).

We can compare equation (1) with this equation: $y = \;f(x)$ -----equation (2)

From equation (1) & (2) we have: \[f(x) = {\csc ^{ - 1}}( - 1)\]

This means we can operate on equation (1) and shift the inverse trigonometric function on the LHS.

$ \Rightarrow \csc y = - 1$----equation (3)

We know that

\[\csc {90^ \circ } = 1\]

Since $\csc y$ is negative in equation (3) so we have to recall the signs of trigonometric functions in all the quadrants which is given in figure given below

From figure (a) we can certainly say that $\sin y$ is positive in 1 st & 2 nd quadrant and similarly $\csc y$ will be positive in the same because both are reciprocal ratios of a triangle i.e. $\sin y = \dfrac{1}{{\csc y}}$ which means it has to be negative in 3 rd and 4 th quadrants.

So the value of $\csc y$ in 3 rd quadrant:

${180^ \circ } + {90^ \circ } = \;{270^ \circ }$

Similarly, the value of $\cot y$ in 4 th quadrant:

${360^\circ } - {90^ \circ } = {270^ \circ }$

From this we can find the value of the equation (3)

\[

\Rightarrow \csc y = - \;1 \\

\Rightarrow \;y = \;{270^ \circ } \\

\]

To convert these degree values into radian values we have to multiply it by $\dfrac{\pi }{{180}}$i.e.

$

y = 270^\circ \\

y = 270 \times \dfrac{\pi }{{180}} \\

y = \dfrac{{3\pi }}{2} \\

$

But we have to find the exact value of the given inverse function for that we check for the range of inverse function cotangent i.e.

$

{\csc ^{ - 1}}(b) = \theta \;;\;where\;\theta \in \;[ - \dfrac{\pi }{2},\dfrac{\pi }{2}] - \{ 0\} $

**From this we can say that**

$

{\csc ^{ - 1}}( - 1) = \dfrac{{3\pi }}{2} = - \dfrac{\pi }{2}{\text{ because}} - \dfrac{\pi }{2} \in [ -

\dfrac{\pi }{2},\dfrac{\pi }{2}] - \{ 0\} \\

\Rightarrow y = - \dfrac{\pi }{2}\; \\

$

$

{\csc ^{ - 1}}( - 1) = \dfrac{{3\pi }}{2} = - \dfrac{\pi }{2}{\text{ because}} - \dfrac{\pi }{2} \in [ -

\dfrac{\pi }{2},\dfrac{\pi }{2}] - \{ 0\} \\

\Rightarrow y = - \dfrac{\pi }{2}\; \\

$

**Note:**We can find angles of trigonometric functions in the manner given above but the important part is to check the range of these inverse trigonometric functions in order to find the exact solution. Changing degree values of angles into radians is not necessary but angles in radians are more preferable at this level.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which places in India experience sunrise first and class 9 social science CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE