Editing Trigonometric functions (section)

==Applications==
{{Main|Uses of trigonometry}}

===Angles and sides of a triangle===
In this section {{mvar|A}}, {{mvar|B}}, {{mvar|C}} denote the three (interior) angles of a triangle, and {{mvar|a}}, {{mvar|b}}, {{mvar|c}} denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.

====Law of sines====
{{Main|Law of sines}}
The law of sines states that for an arbitrary triangle with sides {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} and angles opposite those sides {{mvar|A}}, {{mvar|B}} and {{mvar|C}}:
<math display="block">\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = \frac{2\Delta}{abc},</math>
where {{math|Δ}} is the area of the triangle,
or, equivalently,
<math display="block">\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,</math>
where {{mvar|R}} is the triangle's [[circumscribed circle|circumradius]].

It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in ''[[triangulation]]'', a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

====Law of cosines====
{{Main|Law of cosines}}
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the [[Pythagorean theorem]]:
<math display="block">c^2=a^2+b^2-2ab\cos C,</math>
or equivalently,
<math display="block">\cos C=\frac{a^2+b^2-c^2}{2ab}.</math>

In this formula the angle at {{mvar|C}} is opposite to the side&nbsp;{{mvar|c}}. This theorem can be proved by dividing the triangle into two right ones and using the [[Pythagorean theorem]].

The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.

====Law of tangents====
{{main|Law of tangents}}
The law of tangents says that:
:<math>\frac{\tan \frac{A-B}{2 }}{\tan \frac{A+B}{2 } } = \frac{a-b}{a+b}</math>.

====Law of cotangents====
{{main|Law of cotangents}}
If ''s'' is the triangle's semiperimeter, (''a'' + ''b'' + ''c'')/2, and ''r'' is the radius of the triangle's [[incircle]], then ''rs'' is the triangle's area. Therefore [[Heron's formula]] implies that:

:<math> r = \sqrt{\frac{1}{s} (s-a)(s-b)(s-c)}</math>.

The law of cotangents says that:<ref name="Allen_1976"/>
:<math>\cot{ \frac{A}{2}} = \frac{s-a}{r}</math>
It follows that
:<math>\frac{\cot \dfrac{A}{2}}{s-a}=\frac{\cot \dfrac{B}{2}}{s-b}=\frac{\cot \dfrac{C}{2}}{s-c}=\frac{1}{r}.</math>

===Periodic functions===
[[File:Lissajous curve 5by4.svg|thumb|right|A [[Lissajous curve]], a figure formed with a trigonometry-based function.]]
[[File:Synthesis square.gif|thumb|upright=1.5|right|An animation of the [[additive synthesis]] of a [[Square wave (waveform)|square wave]] with an increasing number of harmonics]]
[[File:Sawtooth Fourier Animation.gif|thumb|upright=1.3|Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental ({{math|1=''k'' = 1}}) have additional nodes. The oscillation seen about the sawtooth when {{mvar|k}} is large is called the [[Gibbs phenomenon]].]]

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe [[simple harmonic motion]], which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of [[uniform circular motion]].

Trigonometric functions also prove to be useful in the study of general [[periodic function]]s. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light [[wave]]s.<ref name="Farlow_1993"/>

Under rather general conditions, a periodic function {{math|1=''f''&hairsp;(''x'')}} can be expressed as a sum of sine waves or cosine waves in a [[Fourier series]].<ref name="Folland_1992"/> Denoting the sine or cosine [[basis functions]] by {{mvar|φ<sub>k</sub>}}, the expansion of the periodic function {{math|1=''f''&hairsp;(''t'')}} takes the form:
<math display="block">f(t) = \sum _{k=1}^\infty c_k \varphi_k(t). </math>

For example, the [[Square wave (waveform)|square wave]] can be written as the [[Fourier series]]
<math display="block"> f_\text{square}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin \big( (2k-1)t \big) \over 2k-1}.</math>

In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a [[sawtooth wave]] are shown underneath.