Question

Inverse circular functions,Principal values of ${\sin }^{-1}x,{cos}^{-1}x,{\tan }^{-1}x$.
${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\frac{x+y}{1-xy}$, $xy<1$
$\pi +{\tan }^{-1}\frac{x+y}{1-xy}$, $xy>1$.
Evaluate the following :
(a) $\sin [\frac{\pi }{3}-{\sin }^{-1}(-\frac{1}{2})]$
(b) $\sin [\frac{\pi }{2}-{\sin }^{-1}(-\frac{\sqrt{3}}{2})]$

Answer 1

(a) $sin[\frac{\pi }{3}-{sin}^{-1}(-\frac{1}{2})]$
$=sin[\frac{\pi }{3}+{sin}^{-1}(-\frac{1}{2})]=sin(\frac{\pi }{3}+\frac{\pi }{6})=sin\frac{\pi }{2}=1$
$\because$ ${sin}^{-1}(-x)={sin}^{-1}x.$.
(b) Just as part (a).
$sin(\frac{\pi }{2}+\frac{\pi }{3})=cos\frac{\pi }{3}=\frac{1}{2}$