# Find the value of cos(sin^-1(sqrt3/2))?

Dec 18, 2016

$\cos \left({\sin}^{-} 1 \left(\frac{\sqrt{3}}{2}\right)\right) = \frac{1}{2}$

#### Explanation:

${\sin}^{-} 1 x$ means an angle whose sine ratio is $x$. If angle is $A$, then it means $\sin A = x$.

Further, although there may be number of values of $A$, for whom sine is $x$ - as all trigonometric ratios have a cycle of $2 \pi$ radians, the range for inverse ratios is limited. While for sine, tangent, cosecant and cotangent range is $\left[- \frac{\pi}{2.} \frac{\pi}{2}\right]$, range for cosine and secant ratios, it is $\left[0 , \pi\right]$.

As $\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$, we have ${\sin}^{-} 1 \left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$ or ${60}^{o}$

and $\cos \left({\sin}^{-} 1 \left(\frac{\sqrt{3}}{2}\right)\right) = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$

Note: It does not matter, whether we write angle in radians or degrees as irrespective of unit used, cosine is a ratio and

even $\cos \left({\sin}^{-} 1 \left(\frac{\sqrt{3}}{2}\right)\right) = \cos {60}^{o} = \frac{1}{2}$