Difference between revisions of "Arccos"

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The function $\mathrm{arccos} \colon [-1,1] \longrightarrow [0,\pi]$ is the [[inverse function]] of the [[cosine]] function.
 
The function $\mathrm{arccos} \colon [-1,1] \longrightarrow [0,\pi]$ is the [[inverse function]] of the [[cosine]] function.
  
[[File:Arccos.png|400px]] [[File:Complex arccos.jpg|400px]]
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<gallery>
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File:Arccos.png|Graph of $\mathrm{arccos}$ on $[-1,1]$.
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File:Complex arccos.jpg|Domain coloring of analytic continuation to $\mathbb{C}$.
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</gallery>
  
 
=Properties=
 
=Properties=
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<strong>Proposition:</strong>  
 
<strong>Proposition:</strong>  
 
$$\dfrac{d}{dz} \mathrm{arccos}(z) = -\dfrac{1}{\sqrt{1-z^2}}$$
 
$$\dfrac{d}{dz} \mathrm{arccos}(z) = -\dfrac{1}{\sqrt{1-z^2}}$$
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Hence substituting back in $y=\mathrm{arccos}(z)$ yields the formula <br />
 
Hence substituting back in $y=\mathrm{arccos}(z)$ yields the formula <br />
 
$$\dfrac{d}{dz} \mathrm{arccos}(z) = -\dfrac{1}{\sin(\mathrm{arccos}(z))} = -\dfrac{1}{\sqrt{1-z^2}}.█$$
 
$$\dfrac{d}{dz} \mathrm{arccos}(z) = -\dfrac{1}{\sin(\mathrm{arccos}(z))} = -\dfrac{1}{\sqrt{1-z^2}}.█$$
 
 
 
 
</div>
 
</div>
 
</div>
 
</div>
  
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<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Proposition:</strong>  
 
<strong>Proposition:</strong>  
 
$$\int \mathrm{arccos}(z) dz = z\mathrm{arccos}(z)-\sqrt{1-z^2}+C$$
 
$$\int \mathrm{arccos}(z) dz = z\mathrm{arccos}(z)-\sqrt{1-z^2}+C$$
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</div>
 
</div>
  
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<div class="toccolours mw-collapsible mw-collapsed">
 
<strong>Proposition:</strong>  
 
<strong>Proposition:</strong>  
 
$$\mathrm{arccos}(z)=\mathrm{arcsec} \left( \dfrac{1}{z} \right)$$
 
$$\mathrm{arccos}(z)=\mathrm{arcsec} \left( \dfrac{1}{z} \right)$$
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</div>
 
</div>
 
</div>
 
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=References=
 
=References=
 
[http://mathworld.wolfram.com/InverseCosine.html  Weisstein, Eric W. "Inverse Cosine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCosine.html]
 
[http://mathworld.wolfram.com/InverseCosine.html  Weisstein, Eric W. "Inverse Cosine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCosine.html]

Revision as of 03:16, 31 October 2014

The function $\mathrm{arccos} \colon [-1,1] \longrightarrow [0,\pi]$ is the inverse function of the cosine function.

Properties

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

Proof: If $y=\mathrm{arccos}(z)$ then $\cos(y)=z$. Now use implicit differentiation with respect to $z$ to get $$-\sin(y)y'=1.$$ If we write $\theta=\mathrm{arccos}(z)$ then the following image shows that $\cos(\mathrm{arccos}(z))=\sqrt{1-z^2}$:

Sin(arccos(z)).png

Hence substituting back in $y=\mathrm{arccos}(z)$ yields the formula
$$\dfrac{d}{dz} \mathrm{arccos}(z) = -\dfrac{1}{\sin(\mathrm{arccos}(z))} = -\dfrac{1}{\sqrt{1-z^2}}.█$$

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

Proof:

Proposition: $$\mathrm{arccos}(z)=\mathrm{arcsec} \left( \dfrac{1}{z} \right)$$

Proof:

References

Weisstein, Eric W. "Inverse Cosine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCosine.html