Question

Find the principal value of \( \sin^{-1}\left(-\frac{1}{2}\right) \).

Hint:

Always check the principal value range before giving answers for inverse trigonometric functions.

Answer 1

To Find the Principal Value of:
sin⁻¹(−1/2)

Step 1: Recall the Principal Value Range
The principal value range of sin⁻¹x is:

[ −π/2 , π/2 ]

So the answer must lie between −π/2 and π/2.

Step 2: Find the Angle
We know:

sin(π/6) = 1/2

Therefore,

sin(−π/6) = −1/2

Since −π/6 lies within the principal value range, it is the required answer.

Final Answer:
sin⁻¹(−1/2) = −π/6