Home
Class 12
MATHS
If x takes negative permissible value th...

If x takes negative permissible value then `sin^(-1)x=`

A

`-cos^(-1)sqrt(1-x^(2))`

B

`cos^(-1)sqrt(x^(2)-1)`

C

`pi-cos^(-1)sqrt(1-x^(2))`

D

`cos^(-1)sqrt(1-x^(2))`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the value of \( \sin^{-1}(x) \) when \( x \) takes negative permissible values. ### Step-by-step Solution: 1. **Understanding the Range of \( \sin^{-1}(x) \)**: The function \( \sin^{-1}(x) \) is defined for \( x \) in the range \([-1, 1]\). When \( x \) is negative, we are looking at values in the interval \([-1, 0)\). 2. **Using the Identity**: We can use the identity for the sine inverse function: \[ \sin^{-1}(-x) = -\sin^{-1}(x) \] This property will help us relate the negative values of \( x \) to their positive counterparts. 3. **Let \( x = -y \)**: Since \( x \) is negative, we can let \( x = -y \), where \( y \) is a positive value in the range \((0, 1]\). Thus, we have: \[ \sin^{-1}(x) = \sin^{-1}(-y) = -\sin^{-1}(y) \] 4. **Finding \( \sin^{-1}(y) \)**: Since \( y \) is in the range \((0, 1]\), \( \sin^{-1}(y) \) will yield a positive angle. Therefore, we can express: \[ \sin^{-1}(x) = -\sin^{-1}(-x) \] 5. **Final Expression**: Thus, the final expression for \( \sin^{-1}(x) \) when \( x \) is negative is: \[ \sin^{-1}(x) = -\sin^{-1}(-x) \] ### Conclusion: If \( x \) takes negative permissible values, then: \[ \sin^{-1}(x) = -\sin^{-1}(-x) \]
Promotional Banner

Similar Questions

Explore conceptually related problems

Write the value of sin(cot^(-1)x) .

Write the value of sin(cot^(-1)x) .

If x + 1/x = 2 , the principal value of sin^(-1) x is

All the permissible value of b ,a=sin(2x-b)if a=0 and x=S_(2) is a subset of (0, pi) are given by

Consider a quadratic expression f (x) =tx^(2) -(2t -1) x+ (5x -1) If f (x) can take both positive and negative values then t must lie in the interval

If the inequality sin^2x+acosx+a^2>1+cosx holds for any x in R , then the largest negative integral value of a is (a) -4 (b) -3 (c) -2 (d) -1

If x >1 , then write the value of sin^(-1)((2x)/(1+x^2)) in terms of tan^(-1)x

If the inequality sin^2x+acosx+a^2>1+cosx holds for any x in R , then the largest negative integral value of a is -4 (b) -3 (c) -2 (d) -1

If the inequality sin^2x+acosx+a^2>1+cosx holds for any x in R , then the largest negative integral value of a is -4 (b) 3 (c) -2 (d) -1

Consider the system of equations sin x cos 2y=(a^(2)-1)^(2)+1, cos x sin 2y = a+1 The number of values of y in [0, 2pi] , when the system has solution for permissible values of a, are