Find the value of y if \(y = {\cos ^{ - 1}}\left( {4{x^3} - 3x} \right

1.	Find the value of y if \(y = {\cos ^{ - 1}}\left( {4{x^3} - 3x} \right) + 3{\sin ^{ - 1}}x\),where x ∈ [-1, 1] ?1. \(\frac{\pi }{2}\)2. \(\frac{{3\pi }}{2}\)3. 4π 4. \( - \frac{\pi }{2}\)
Answer» Correct Answer - Option 2 : \(\frac{{3\pi }}{2}\) Concept: It is known that, \({\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \frac{\pi }{2},x \in \left[ { - 1,1} \right]\) Calculation: \(y = {\cos ^{ - 1}}\left( {4{x^3} - 3x} \right) + 3{\sin ^{ - 1}}x\)---------(1) Take \({\cos ^{ - 1}}\left( {4{x^3} - 3x} \right)\) Let us substitute x = cos θ in \({\cos ^{ - 1}}\left( {4{x^3} - 3x} \right)\) \({\cos ^{ - 1}}\left( {4{x^3} - 3x} \right) = {\cos ^{ - 1}}\left( {4{{\cos }^3}θ - 3\cos θ } \right)\) \(= {\cos ^{ - 1}}\left( {\cos 3θ } \right)\) \(= 3θ \) \(= 3{\cos ^{ - 1}}x\) Now substitute the value of \({\cos ^{ - 1}}\left( {4{x^3} - 3x} \right)\) in equation (1) \(y = {\cos ^{ - 1}}\left( {4{x^3} - 3x} \right) + 3{\sin ^{ - 1}}x\) \( = 3{\cos ^{ - 1}}x + 3{\sin ^{ - 1}}x\) As we know that, \({\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \frac{\pi }{2},x \in \left[ { - 1,1} \right]\) \( y = \frac{3\pi}{2}\)

Find the value of y if \(y = {\cos ^{ - 1}}\left( {4{x^3} - 3x} \right) + 3{\sin ^{ - 1}}x\),where x ∈ [-1, 1] ?1. \(\frac{\pi }{2}\)2. \(\frac{{3\pi }}{2}\)3. 4π 4. \( - \frac{\pi }{2}\)

Answer» Correct Answer - Option 2 : \(\frac{{3\pi }}{2}\)

Concept:

It is known that, \({\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \frac{\pi }{2},x \in \left[ { - 1,1} \right]\)

Calculation:

\(y = {\cos ^{ - 1}}\left( {4{x^3} - 3x} \right) + 3{\sin ^{ - 1}}x\)---------(1)

Take \({\cos ^{ - 1}}\left( {4{x^3} - 3x} \right)\)

Let us substitute x = cos θ in \({\cos ^{ - 1}}\left( {4{x^3} - 3x} \right)\)

\({\cos ^{ - 1}}\left( {4{x^3} - 3x} \right) = {\cos ^{ - 1}}\left( {4{{\cos }^3}θ - 3\cos θ } \right)\)

\(= {\cos ^{ - 1}}\left( {\cos 3θ } \right)\)

\(= 3θ \)

\(= 3{\cos ^{ - 1}}x\)

Now substitute the value of \({\cos ^{ - 1}}\left( {4{x^3} - 3x} \right)\) in equation (1)

\(y = {\cos ^{ - 1}}\left( {4{x^3} - 3x} \right) + 3{\sin ^{ - 1}}x\)

\( = 3{\cos ^{ - 1}}x + 3{\sin ^{ - 1}}x\)

As we know that, \({\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \frac{\pi }{2},x \in \left[ { - 1,1} \right]\)

\( y = \frac{3\pi}{2}\)