Question

4. Find the exact value of tan(sin^-1(-1/2))

Answer 1

$tan(sin^{-1}(-1/2)) = sin(sin^{-1}(-1/2))/cos(sin^{-1}(-1/2));$

By the definition of inverse function( $sin(sin^{-1}(x)) = x)$ :

$tan(sin^{−1} (−1/2)) = -1/2*cos(sin^{-1}(-1/2));$

By the Pythagorean identity:

$cos(sin^{-1}(-1/2)) = \sqrt{1-sin^2(sin^{-1}(-1/2))} = \sqrt3/2$;

$tan(sin^{-1}(-1/2)) = -1/\sqrt3 = -\sqrt3/3$