Angles and sides of a triangle

In this section $A$ , $B$ , $C$ denote the three (interior) angles of a triangle, and $a$ , $b$ , $c$ 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

The law of sines states that for an arbitrary triangle with sides $a$ , $b$ , and $c$ and angles opposite those sides $A$ , $B$ and $C$ : ${\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},$ where $Δ$ is the area of the triangle, or, equivalently, ${\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,$ where $R$ is the triangle's 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

The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem: $c^{2}=a^{2}+b^{2}-2ab\cos C,$ or equivalently, $\cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.$

In this formula the angle at $C$ is opposite to the side $c$ . 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

The law of tangents says that:

{\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}

.

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:

r={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}

.

The law of cotangents says that:^[27]

\cot {\frac {A}{2}}={\frac {s-a}{r}}

It follows that

{\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{r}}.

Periodic functions

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 functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.^[28]

Under rather general conditions, a periodic function $f (x)$ can be expressed as a sum of sine waves or cosine waves in a Fourier series.^[29] Denoting the sine or cosine basis functions by $φ k$ , the expansion of the periodic function $f (t)$ takes the form: $f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).$

For example, the square wave can be written as the Fourier series $f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.$

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.