Arcsin

The $\mathrm{arcsin}$ function is the inverse function of the sine function.
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Properties

Proposition: $$\dfrac{d}{dz} \mathrm{arcsin(z)} = \dfrac{1}{1-z^2}$$

Proof: █

Proposition: $$\int \mathrm{arcsin}(z) dz = \sqrt{1-z^2}+z\mathrm{arcsin}(z)+C$$

Proof: █

Proposition: $$\mathrm{arcsin}(z) = \mathrm{arccsc}\left( \dfrac{1}{z} \right)$$

Proof: █

Proposition: $$\mathrm{arcsin}(z)=\sum_{k=0}^{\infty} \dfrac{\left(\frac{1}{2} \right)_n}{(2n+1)n!}x^{2n+1}$$

Proof: █

References

• Weisstein, Eric W. "Inverse Sine." From MathWorld--A Wolfram Web Resource.