Difference between revisions of "Arcsin"
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(Created page with "The $\mathrm{arcsin}$ function is the inverse function of the sine function. <br /> 500px") |
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The $\mathrm{arcsin}$ function is the inverse function of the [[sine]] function. <br /> | The $\mathrm{arcsin}$ function is the inverse function of the [[sine]] function. <br /> | ||
[[File:Arcsin.png|500px]] | [[File:Arcsin.png|500px]] | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Proposition:</strong> | ||
| + | $$\dfrac{d}{dz} \mathrm{arcsin(z)} = \dfrac{1}{1-z^2}$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Proposition:</strong> | ||
| + | $$\int \mathrm{arcsin}(z) dz = \sqrt{1-z^2}+z\mathrm{arcsin}(z)+C$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Proposition:</strong> | ||
| + | $$\mathrm{arcsin}(z) = \mathrm{arccsc}\left( \dfrac{1}{z} \right)$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Proposition:</strong> | ||
| + | $$\mathrm{arcsin}(z)=\sum_{k=0}^{\infty} \dfrac{\left(\frac{1}{2} \right)_n}{(2n+1)n!}x^{2n+1}$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | =References= | ||
| + | *[http://mathworld.wolfram.com/InverseSine.html Inverse sine on Mathworld] | ||
Revision as of 01:25, 19 October 2014
The $\mathrm{arcsin}$ function is the inverse function of the sine function.
500px
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: █
