# How do you find the value of sin (arc sec(x))?

Nov 23, 2016

$\sin \left(a r c \sec \left(x\right)\right) = \frac{\sqrt{{x}^{2} - 1}}{x}$

#### Explanation:

$a r c \sec x$ means the angle whose secant ratio is $x$ i.e. if $a r c \sec \left(x\right) = \theta$, we have $\sec \theta = x$.

As $\sec \theta = x$, we have $\cos \theta = \frac{1}{x}$

and $\sin x = \sqrt{1 - \frac{1}{x} ^ 2} = \frac{\sqrt{{x}^{2} - 1}}{x}$

Hence, $\sin \left(a r c \sec \left(x\right)\right) = \sin \theta = \frac{\sqrt{{x}^{2} - 1}}{x}$