nLab trigonometric function

Idea

A trigonometric function is one derived from the basic functions of trigonometry, $\cos$, $\sin$ (cosine and sine), which themselves are the standard coordinate functions (equivalently: product projections $pr_i$) of the Cartesian space $\mathbb{R}^2$, restricted to the unit circle:

$$\array{ & && \mathbb{R} \\ & & {}^{\mathllap{cos}}\nearrow & \big\uparrow{}^{ \mathrlap{pr_1} } \\ \mathbb{R} \overset{exp}{\longrightarrow} & S^1 &\hookrightarrow& \mathbb{R}^2 &\simeq& \mathbb{R} \times \mathbb{R} \\ & & {}_{\mathllap{sin}}\searrow & \big\downarrow{}^{ \mathrlap{pr_2} } \\ & && \mathbb{R} } \,,$$

or rather the result of composing these restrictions with an arc length parametrization $\mathbb{R} \overset{\exp}{\to} S^1$. They are also called circular functions.

In elementary mathematics, there are six traditional trigonometric functions;

1. sine$\;$ $sin$

2. cosine$\;$ $cos$

3. tangent$\;$ $\tan = \frac{\sin}{\cos}$,

4. cotangent$\;$ $\cot = \frac{\cos}{\sin}$,

5. secant?$\;$ $\sec = \frac1{\cos}$,

6. 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”).

These six functions figure prominently in Euclidean geometry where the angles of a triangle sum to arc length $\pi$.

There are more elaborate offshoots such as spherical trigonometry? (see elliptic geometry) which was historically important for terrestrial navigation. Moreover, there are the related hyperbolic functions (see hyperbolic geometry) which result from a projective change of conic section (from a circle to a hyperbola).

Definition

The most useful modern approach to cos and sin comes from taking Euler's formula as a working definition:

Let

$$\exp \;\colon\; \mathbb{C} \longrightarrow \mathbb{C}^\ast$$

be the complex exponential function, defined by the exponential power series formula

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

These satisfy the following 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.

Related concepts

• inverse trigonometric function

• trigonometry

• trigonometric identity

• hyperbolic function

• elliptic function

References

• Wikipedia, Trigonometric function

• Springer eom: V.I. Bityutskov, Trigonometric functions