nLab
trigonometric function

Contents

Idea

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

In elementary mathematics, there are six traditional trigonometric functions; in addition to sin\sin and cos\cos they are the tangent tan=sincos\tan = \frac{\sin}{\cos}, the cotangent cot=cossin\cot = \frac{\cos}{\sin}, the secant sec=1cos\sec = \frac1{\cos}, and the 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”). They figure heavily in Euclidean geometry where the angles of a triangle sum to 180 180^\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\cos, sin\sin comes from taking Euler's formula as a working definition. Let exp: *\exp: \mathbb{C} \to \mathbb{C}^\ast be the complex exponential function, defined by the power series formula

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

As is well-known, we have some 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 September 15, 2015 at 14:34:39. See the history of this page for a list of all contributions to it.