# Contents

## Idea

A trigonometric function is one derived from the basic trigonometric functions $\cos$, $\sin$ (cosine and sine), which are the coordinate (or product projection) functions restricted to the unit circle $S^1 \hookrightarrow \mathbb{R} \times \mathbb{R}$, or rather the result of composing these restrictions with an arc length? parametrization $\mathbb{R} \to S^1$. They are also called circular functions.

In elementary mathematics, there are six traditional trigonometric functions; in addition to $\sin$ and $\cos$ they are the tangent $\tan = \frac{\sin}{\cos}$, the cotangent $\cot = \frac{\cos}{\sin}$, the secant $\sec = \frac1{\cos}$, and the cosecant $\csc = \frac1{\sin}$. The early view was that these functions measured the six possible ratios of side lengths of right triangles (as a basic case in terms of which other triangles can be analyzed; “trigonometry” = “triangle measure”). They figure heavily in Euclidean geometry where the angles of a triangle sum to $180^\circ$; there are more elaborate offshoots such as spherical trigonometry? which was historically important for terrestrial navigation, and then there are the related hyperbolic functions which result from a projective change of conic section (from a circle to a hyperbola).

## Definition

The most useful modern approach to $\cos$, $\sin$ comes from taking Euler's formula as a working definition. Let $\exp: \mathbb{C} \to \mathbb{C}^\ast$ be the complex exponential function, defined by the power series formula

$\exp(z) = \sum_{n \geq 0} \frac{z^n}{n!}.$

As is well-known, we have some fundamental exponential identities:

• $\exp(z+w) = \exp(z) \cdot \exp(w)$ (homomorphism from an additive group to a multiplicative group),

• $\widebar{\exp(z)} = \exp(\widebar{z})$ (because complex conjugation $\widebar{(-)}$ is a continuous field automorphism).

As a result, if $z + \widebar{z} = 0$ (i.e., if $z$ is purely imaginary: $z = i t$ for some $t \in \mathbb{R}$), we have

$\exp(z) \cdot \widebar{\exp(z)} = 1$

so that $w = \exp(z)$ lies on the unit circle defined by $w\widebar{w} = 1$.

###### Definition

The cosine function $\cos: \mathbb{R} \to \mathbb{R}$ is defined by $\cos(t) = Re(\exp(i t))$ (real part); the sine function $\sin: \mathbb{R} \to \mathbb{R}$ is defined by $\sin(t) = Im(\exp(i t))$ (imaginary part). In other words: $\exp(i t) = \cos(t) + i\sin(t)$ (Euler).

This implies that $\cos(t) = \frac1{2}(\exp(i t) + \exp(-i t))$ and $\sin(t) = \frac1{2 i}(\exp(i t) - \exp(-i t))$; these equations suggest the simple analytic continuation of $\cos$ and $\sin$ to functions $\mathbb{C} \to \mathbb{C}$ on the entire complex plane.

## Properties

More or less immediate consequences of the definition include

• $(\cos t)^2 + (\sin t)^2 = 1$ (“Pythagorean theorem”), since $\exp(i t)$ lies on the unit circle. This is traditionally written as $\cos^2(t) + \sin^2(t) = 1$;

• $\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)$ and $\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)$ (“addition formulas”), by taking real and imaginary parts of the identity that says $\exp$ is a homomorphism;

• $(\sin)' = \cos$ and $(\cos)' = -\sin$ (by differentiating $\exp(i t)$); the connection with the arc length parametrization of the unit circle is that the derivative of the position vector $p'(t) = (\exp(i t))'$ is a velocity vector $i\exp(i t)$ of unit length;

• $\cos(x + 2\pi) = \cos(x)$ and $\sin(x+2\pi) = \sin(x)$ (“periodicity”), according to the modern definition of pi;

• $\cos(x) = \sum_{n \geq 0} (-1)^n \frac{x^{2 n}}{(2 n)!}$ and $\sin(x) = \sum_{n \geq 0} (-1)^n \frac{x^{2 n + 1}}{(2 n + 1)!}$, by exploiting the power series representation of the exponential function.

These name but a few of many trigonometric identities, facility in which can serve as a modern-day shibboleth or barrier of passage in high school or lower-level undergraduate courses in mathematics. They seem also to be popular in mathematics education in India and appear regularly in entrance examinations there. But the ones listed above are the most fundamental.

There is no question that the trigonometric functions are rich in significance; for example; various representations of the tangent, cotangent, secant, etc. appear in enumerative combinatorics (as in the problem of counting alternating permutations), representations of Bernoulli numbers, and so on.

## References

Last revised on September 15, 2015 at 14:34:39. See the history of this page for a list of all contributions to it.