Difference between revisions of "Cosine"

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The cosine function, $\cos \colon \mathbb{R} \rightarrow \mathbb{R}$ is the unique solution of the second order initial value problem
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The cosine function, $\cos \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula
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$$y''=-y;y(0)=1,y'(0)=0.$$
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$$\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2},$$
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where $e^z$ is the [[exponential function]].
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
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File:Cosine.png|Graph of $\cos$ on $\mathbb{R}$.
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File:Cosineplot.png|Graph of $\cos$ on $[-2\pi,2\pi]$.
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File:Complex cos.jpg|Domain coloring of analytic continuation of $\cos$ to $\mathbb{C}$.
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File:Complexcosineplot.png|[[Domain coloring]] of $\cos$.
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File:Trig Functions Diagram.svg|Trig functions diagram using the unit circle.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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[[Derivative of cosine]]<br />
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<strong>Proposition:</strong> $$\cos(x) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k x^{2k}}{(2k)!}$$
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[[Taylor series of cosine]]<br />
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<div class="mw-collapsible-content">
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[[Weierstrass factorization of cosine]]<br />
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<strong>Proof:</strong> █
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[[Beta in terms of sine and cosine]]<br />
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</div>
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[[Relationship between cosine and hypergeometric 0F1]]<br />
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</div>
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[[Relationship between spherical Bessel y and cosine]]<br />
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[[Relationship between cosh and cos]]<br />
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[[Relationship between cos and cosh]]<br />
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[[Relationship between cosine, Gudermannian, and sech]]<br />
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[[Relationship between sech, inverse Gudermannian, and cos]]<br />
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[[2cos(mt)cos(nt)=cos((m+n)t)+cos((m-n)t)]]<br />
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[[Orthogonality relation for cosine on (0,pi)]]<br />
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=See Also=
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[[Arccos]] <br />
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[[Cosh]] <br />
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[[Arccosh]] <br />
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=References=
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Sine|next=Tangent}}: 4.3.2
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{{:Trigonometric functions footer}}
  
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<div class="toccolours mw-collapsible mw-collapsed">
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[[Category:SpecialFunction]]
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<strong>Proposition:</strong> $$\cos(x) = \prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 

Latest revision as of 22:09, 19 December 2017

The cosine function, $\cos \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula $$\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2},$$ where $e^z$ is the exponential function.

  • Graph of $\cos$ on $[-2\pi,2\pi]$.

  • Domain coloring of $\cos$.

  • Trig functions diagram using the unit circle.

Properties

Derivative of cosine
Taylor series of cosine
Weierstrass factorization of cosine
Beta in terms of sine and cosine
Relationship between cosine and hypergeometric 0F1
Relationship between spherical Bessel y and cosine
Relationship between cosh and cos
Relationship between cos and cosh
Relationship between cosine, Gudermannian, and sech
Relationship between sech, inverse Gudermannian, and cos
2cos(mt)cos(nt)=cos((m+n)t)+cos((m-n)t)
Orthogonality relation for cosine on (0,pi)

See Also

Arccos
Cosh
Arccosh

References

Trigonometric functions

Sine

Cosine

Tangent

Cosecant

Secant

Cotangent
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