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:

\(\ds \map \arctan {\dfrac x {\sqrt {1 - x^2} } }\)

\(=\)

\(\ds \map \arctan {\dfrac {\sin \theta} {\sqrt {1 - \sin^2 \theta} } }\)

\(\ds \)

\(=\)

\(\ds \map \arctan {\dfrac {\sin \theta} {\sqrt {\cos^2 \theta} } }\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds \map \arctan {\tan \theta}\)

Definition of Real Tangent Function

\(\ds \)

\(=\)

\(\ds \theta\)

Definition of Real Arctangent

\(\ds \)

\(=\)

\(\ds \arcsin x\)

$\blacksquare$

Also see

• Arccosine in terms of Arctangent
• Arctangent in terms of Arcsine