The Physics Travel Guide

Trigonometric Functions

Best explanations

Intuitive

Picture a point moving around a circle of radius 1 at constant speed, like the tip of a clock hand. At every moment the point has a horizontal position and a vertical position. As the hand turns, the horizontal position swings back and forth between minus one and plus one, and so does the vertical one, a quarter turn behind. These two positions, as functions of the angle the hand has turned through, are the cosine (horizontal) and the sine (vertical).

The same two numbers describe right triangles. Drop a line from the moving point straight down to the horizontal axis: you get a right triangle whose longest side is the clock hand. The sine is the height of that triangle divided by its longest side, and the cosine is its base divided by its longest side. Because all right triangles with the same angle have the same shape, these ratios depend only on the angle. That is what makes them useful for surveying and navigation: measure an angle and one length, and the other lengths follow.

In physics the circle picture matters more. The simplest oscillations, such as a mass on a spring, a pendulum with small swings or the voltage in a power outlet, move like the shadow of a point going around a circle. This is why sine and cosine are the basic shapes of oscillations and waves. More complicated periodic signals, such as the sound of an instrument, can be built from sums of them.

Concrete

Definitions

In a right triangle with an angle θ\theta, opposite side aa, adjacent side bb and hypotenuse cc,

sin⁡θ=ac,cos⁡θ=bc,tan⁡θ=sin⁡θcos⁡θ=ab.\sin\theta = \frac{a}{c} , \qquad \cos\theta = \frac{b}{c} , \qquad \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{a}{b} .

This only covers 0<θ<π/20 < \theta < \pi/2. The general definition uses the unit circle: the point reached by turning an angle θ\theta counterclockwise from the positive xx-axis is (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta). Angles are measured in radians, so a full turn is 2π2\pi (see Pi). Both functions are periodic with period 2π2\pi, and cos⁡θ=sin⁡(θ+π/2)\cos\theta = \sin(\theta + \pi/2).

Identities

Since the point lies on the unit circle, the Pythagorean theorem gives

sin⁡2θ+cos⁡2θ=1.\sin^2\theta + \cos^2\theta = 1 .

The addition theorems

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β,cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\begin{aligned} \sin(\alpha + \beta) &= \sin\alpha\cos\beta + \cos\alpha\sin\beta , \\ \cos(\alpha + \beta) &= \cos\alpha\cos\beta - \sin\alpha\sin\beta \end{aligned}

follow from composing two rotations. Most other identities, such as sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta, are special cases. With Euler's formula eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta they reduce to ei(α+β)=eiαeiβe^{i(\alpha+\beta)} = e^{i\alpha}e^{i\beta}.

Calculus

With θ\theta in radians,

ddθsin⁡θ=cos⁡θ,ddθcos⁡θ=−sin⁡θ,\frac{d}{d\theta}\sin\theta = \cos\theta , \qquad \frac{d}{d\theta}\cos\theta = -\sin\theta ,

and for small angles sin⁡θ≈θ\sin\theta \approx \theta, cos⁡θ≈1−θ2/2\cos\theta \approx 1 - \theta^2/2. Differentiating twice gives d2dθ2sin⁡θ=−sin⁡θ\frac{d^2}{d\theta^2}\sin\theta = -\sin\theta, which is the reason these functions solve the harmonic oscillator.

Worked example: harmonic oscillator

A mass mm on a spring with constant kk obeys mx¨=−kxm\ddot x = -kx, that is x¨=−ω2x\ddot x = -\omega^2 x with ω=k/m\omega = \sqrt{k/m}. The general solution is

x(t)=Acos⁡(ωt+φ),x(t) = A\cos(\omega t + \varphi) ,

with amplitude AA and phase φ\varphi fixed by the initial position and velocity. The period is T=2π/ωT = 2\pi/\omega. A pendulum at small amplitude reduces to this case through sin⁡θ≈θ\sin\theta \approx \theta.

Vectors and waves

A vector of length rr at angle θ\theta to the xx-axis has components (rcos⁡θ,rsin⁡θ)(r\cos\theta, r\sin\theta), which is how forces are decomposed on an inclined plane. A wave travelling in the xx direction is cos⁡(kx−ωt)\cos(kx - \omega t) with wave number k=2π/λk = 2\pi/\lambda.

Abstract

Analytically, sine and cosine are defined by their power series

sin⁡z=∑n=0∞(−1)nz2n+1(2n+1)!,cos⁡z=∑n=0∞(−1)nz2n(2n)!,\sin z = \sum_{n=0}^\infty \frac{(-1)^n z^{2n+1}}{(2n+1)!} , \qquad \cos z = \sum_{n=0}^\infty \frac{(-1)^n z^{2n}}{(2n)!} ,

which converge on all of C\mathbb{C}, so both are entire functions. Equivalently, cos⁡z=12(eiz+e−iz)\cos z = \frac{1}{2}(e^{iz} + e^{-iz}) and sin⁡z=12i(eiz−e−iz)\sin z = \frac{1}{2i}(e^{iz} - e^{-iz}). On the real line they are bounded by 1; on C\mathbb{C} they are not, for example cos⁡(iy)=cosh⁡y\cos(iy) = \cosh y. Another characterization: sin⁡\sin is the unique solution of f′′=−ff'' = -f with f(0)=0f(0) = 0, f′(0)=1f'(0) = 1, and cos⁡\cos the one with f(0)=1f(0) = 1, f′(0)=0f'(0) = 0.

Group theory

The map θ↦eiθ\theta \mapsto e^{i\theta} is a homomorphism from (R,+)(\mathbb{R}, +) onto the circle group U(1)U(1) with kernel 2πZ2\pi\mathbb{Z}. Identifying U(1)≅SO(2)U(1) \cong SO(2), it reads

θ↦R(θ)=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)=exp⁡ ⁣(θ(0−110)),\theta \mapsto R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} = \exp\!\left( \theta \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \right) ,

and R(α)R(β)=R(α+β)R(\alpha)R(\beta) = R(\alpha+\beta) contains both addition theorems. Sine and cosine are thus the matrix elements of the defining representation of the one-parameter group SO(2)SO(2). The irreducible unitary representations of U(1)U(1) are the characters einθe^{in\theta}, n∈Zn \in \mathbb{Z}, and by the Peter-Weyl theorem they form an orthonormal basis of L2(S1)L^2(S^1) (up to normalization). This is the completeness of Fourier series, see the Fourier transform.

Hyperbolic functions

Replacing the Euclidean form x2+y2x^2 + y^2 by the Lorentzian form t2−x2t^2 - x^2 replaces SO(2)SO(2) by SO(1,1)SO(1,1), and the rotation matrix by the boost

(cosh⁡ηsinh⁡ηsinh⁡ηcosh⁡η),cosh⁡2η−sinh⁡2η=1,\begin{pmatrix} \cosh\eta & \sinh\eta \\ \sinh\eta & \cosh\eta \end{pmatrix} , \qquad \cosh^2\eta - \sinh^2\eta = 1 ,

with rapidity η\eta. Rapidities add under collinear boosts the same way angles add under rotations. The relations cos⁡(iz)=cosh⁡z\cos(iz) = \cosh z and sin⁡(iz)=isinh⁡z\sin(iz) = i\sinh z connect the two cases. The same continuation t→−iτt \to -i\tau underlies the Wick rotation.

Why is it interesting?

Imagine Bob The Alien visits Earth to study our species.

Without new words, humans are hard to describe: “There’s a sphere at the top, which gets scratched occasionally” or “Two elongated cylinders appear to provide locomotion”.

After creating specific terms for anatomy, Bob might jot down typical body proportions:

  • The armspan (fingertip to fingertip) is approximately the height
  • A head is 5 eye-widths wide
  • Adults are 8 head-heights tall

How is this helpful?

Well, when Bob finds a jacket, he can pick it up, stretch out the arms, and estimate the owner’s height. And head size. And eye width. One fact is linked to a variety of conclusions.

Even better, human biology explains human thinking. Tables have legs, organizations have heads, crime bosses have muscle. Our biology offers ready-made analogies that appear in man-made creations.

Now the plot twist: you are Bob the alien, studying creatures in math-land!

Generic words like “triangle” aren’t overly useful. But labeling sine, cosine, and hypotenuse helps us notice deeper connections. And scholars might study haversine, exsecant and gamsin, like biologists who find a link between your tibia and clavicle.

And because triangles show up in circles…

…and circles appear in cycles, our triangle terminology helps describe repeating patterns!

Trig is the anatomy book for “math-made” objects. If we can find a metaphorical triangle, we’ll get an armada of conclusions for free. How To Learn Trigonometry Intuitively by Kalid Azad

Leads to: Euler's Formula, Pi / $\pi$.