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Need solution for RD Sharma Maths Class 12 Chapter 3 Inverse Trigonometric Functions Excercise Fill in the Blanks Question 3

Answers (1)

Answer:

                \left [ \frac{\pi}{4} ,\frac{3\pi}{4}\right ]

Hint:

You must know the value of trigonometric and inverse trigonometric function.

Given:

                Range of \sin^{-1}x+\cos^{-1}x+\tan^{-1}x

Solution:

                \sin^{-1}x+\cos^{-1}x+\tan^{-1}x=f\left ( x \right )

                \sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}     [Inverse trigonometric function]

So, we get

\Rightarrow         \frac{\pi}{2}+\tan^{-1}x=f\left ( x \right )

Now, most of us apply the range of \tan^{-1}x as  \left [ \frac{-\pi}{2},\frac{\pi}{2} \right ]

So, range of  f\left ( x \right )=\left [ 0,\pi \right ]

But, we know that domain of f\left ( x \right ) is \left [ -1,1 \right ]

Hence, term \tan^{-1}x does not hold this value\frac{-\pi}{2}  and \frac{\pi}{2}

So new range becomes,

\Rightarrow f\left ( -1 \right )=\frac{\pi}{2}-\frac{\pi}{4}=\frac{\pi}{4}        

\Rightarrow f\left ( 1 \right )=\frac{\pi}{2}+\frac{\pi}{4}=\frac{3\pi}{4}        

So, we get  f\left ( x \right )\epsilon \left [ \frac{\pi}{4},\frac{3\pi}{4} \right ]

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