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Question

Mathematics Question on Inverse Trigonometric Functions

Find the principal value of sin-1(−12)\Big(\frac {-1}{2}\Big)

Answer

Let sin-1(−12)=y\Big(\frac{-1}{2}\Big)=y.
Then sin y = −12=−sin(π6)=sin(−π6)- \frac {1}{2}=-sin\bigg( \frac {\pi}{6}\bigg)=sin \bigg(\frac{-\pi}{6}\bigg).
We know that the range of the principal value branch of sin-1
is [−π2,π2]\bigg[\frac {-\pi}{2}, \frac {\pi}{2}\bigg]and sin (−π6)=−12\Big(\frac{-\pi}{6}\Big)=-\frac{1}{2}
Therefore, the principal value of sin-1 (−12)is−π6\Big(\frac{-1}{2}\Big) is \frac {-\pi}{6}