Arcsine in terms of Arctangent

Theorem

 * $\arcsin x = \map \arctan {\dfrac x {\sqrt {1 - x^2} } }$

where $x$ is a real number with $-1 < x < 1$.

Proof
Let:
 * $\theta = \arcsin x$

Then by the definition of arcsine:
 * $x = \sin \theta$

and:
 * $-\dfrac \pi 2 < \theta < \dfrac \pi 2$

Then:

Also see

 * Arccosine in terms of Arctangent
 * Arctangent in terms of Arcsine