nLab
trigonometric function

Contents

Contents

Idea

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

cos pr 1 exp S 1 2 × sin pr 2 , \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 expS 1\mathbb{R} \overset{\exp}{\to} S^1. They are also called circular functions.

In elementary mathematics, there are six traditional trigonometric functions;

  1. sine\; sinsin

  2. cosine\; coscos

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

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

  5. secant?\; sec=1cos\sec = \frac1{\cos},

  6. cosecant?\; csc=1sin\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 π\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: * \exp \;\colon\; \mathbb{C} \longrightarrow \mathbb{C}^\ast

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

exp(z) n0z nn!. \exp(z) \;\coloneqq\; \sum_{n \geq 0} \frac{z^n}{n!} \,.

These satisfy the following fundamental exponential identities:

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

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

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

Definition

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

This implies that cos(t)=12(exp(it)+exp(it))\cos(t) = \frac1{2}(\exp(i t) + \exp(-i t)) and sin(t)=12i(exp(it)exp(it))\sin(t) = \frac1{2 i}(\exp(i t) - \exp(-i t)); these equations suggest the simple analytic continuation of cos\cos and sin\sin to functions \mathbb{C} \to \mathbb{C} on the entire complex plane.

Properties

More or less immediate consequences of the definition include

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

  • cos(x+y)=cos(x)cos(y)sin(x)sin(y)\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y) and sin(x+y)=sin(x)cos(y)+cos(x)sin(y)\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\exp is a homomorphism;

  • (sin)=cos(\sin)' = \cos and (cos)=sin(\cos)' = -\sin (by differentiating exp(it)\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(it))p'(t) = (\exp(i t))' is a velocity vector iexp(it)i\exp(i t) of unit length;

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

  • cos(x)= n0(1) nx 2n(2n)!\cos(x) = \sum_{n \geq 0} (-1)^n \frac{x^{2 n}}{(2 n)!} and sin(x)= n0(1) nx 2n+1(2n+1)!\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 May 10, 2019 at 11:03:01. See the history of this page for a list of all contributions to it.