Trigonometry & The Unit Circle

The circle that draws a wave

Junior level — plain language, no maths

Draw a circle and mark a dot on its edge. Now spin the dot slowly around, and watch only its height - how far above or below the middle it is. Something magical appears: as the dot goes round and round, its height traces out a smooth, endlessly repeating wave. That wave has a name, the sine wave, and it was hiding inside the humble circle all along.

The two numbers that pin down the dot are the building blocks of trigonometry. Its height is the sine of the angle; how far left or right it is, is the cosine. As the angle grows from 0° all the way to 360° and starts over, sine and cosine slide smoothly between +1 and -1. That's all they really are: the vertical and horizontal position of a point going around a circle of radius 1 - the unit circle.

This isn't just maths-class trivia. That same circle-into-wave pattern sits behind almost everything that repeats: the sound of a musical note, the swing of a pendulum, the tides, the alternating current in your walls, the bob of a boat. Anything that cycles can be written with sines and cosines - which is why this one little picture turns up in physics, engineering, music and computer graphics everywhere you look.

The unit circle, and sine, cosine and tangent

Student level — the core equations

Put a circle of radius 1 at the origin and take a point on it at angle \(\theta\) (measured anticlockwise from the positive x-axis). By definition that point's coordinates are \((\cos\theta,\ \sin\theta)\). So cosine is the x-coordinate and sine is the y-coordinate - full stop. Everything else in trigonometry unfolds from that. And because the radius is 1, Pythagoras instantly gives the identity \(\sin^2\theta + \cos^2\theta = 1\).

In a right-angled triangle the same functions are ratios of sides: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent = \(\sin\theta/\cos\theta\). The unit circle and the triangle are the same idea - the triangle is just the little right triangle formed by the point, the centre and the x-axis. Tangent measures the slope of the radius line, and shoots off to infinity at 90°, where that line stands vertical.

Because the point returns to where it started every full turn, all three functions are periodic, repeating every 360° (or \(2\pi\) radians). Radians, where a whole circle is \(2\pi\), are the natural unit: the angle in radians is literally the arc length around a unit circle. Speed up the rotation and you change the wave's frequency; scale the circle and you change its amplitude - the two knobs behind every wave in nature.

From triangles to Taylor series, waves and rotations

Scholar level — full mathematical depth

01Radians and calculus

Sine and cosine become clean only in radians: \(\tfrac{d}{dx}\sin x = \cos x\) and \(\tfrac{d}{dx}\cos x = -\sin x\). Each is the other's derivative (up to a sign), so together they satisfy \(y'' = -y\) - the differential equation of every oscillation, from a spring to an LC circuit.

02Taylor series and Euler's formula

Both have infinite polynomial (Taylor) expansions, and those knit together with the exponential through Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\). Set \(\theta = \pi\) and you get \(e^{i\pi} + 1 = 0\), five fundamental constants in a single line. Complex exponentials are the physicist's favourite disguise for waves.

03Fourier: everything is sines

Any repeating signal, however jagged, is a sum of sines and cosines of different frequencies - a Fourier series. This is why the unit circle quietly underlies audio compression, image formats, signal processing and the mathematics of quantum mechanics.

04Rotations and beyond

Sine and cosine are the entries of the 2D rotation matrix, so they drive all of computer graphics, robotics and orbital mechanics. The same pair generalizes to the hyperbolic \(\sinh\) and \(\cosh\) - where the circle becomes a hyperbola - which describe hanging cables and the geometry of special relativity.