MISSION · Geometry
The Unit Circle
How a point going round a circle becomes a wave
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The idea in one sentence
Put a point on a circle of radius 1 and spin it: its horizontal position is \(\cos \theta\), its vertical position is \(\sin \theta\), and now those functions work for any angle at all.
Why the definition has to change
The triangle definitions run out fast. A triangle can’t have an angle of \(120°\) alongside a right angle, and \(400°\) means nothing to a triangle. Yet sine and cosine are needed well past \(90°\) — for waves, rotation, orbits, and alternating current.
So the definition is rebuilt on a circle. Take the unit circle, \(x^2 + y^2 = 1\), measure the angle \(\theta\) anticlockwise from the positive \(x\)-axis, and define:
\[\cos \theta = x \qquad \sin \theta = y\]For angles under \(90°\) this agrees exactly with the triangle version — the radius is the hypotenuse, and since it equals 1, \(\frac{\text{opp}}{\text{hyp}}\) is just the height. Nothing is contradicted; the definition is extended.
The signs take care of themselves
Past \(90°\) the point moves into quadrants where the coordinates go negative, and the functions follow:
| Quadrant | \(\cos\) | \(\sin\) |
|---|---|---|
| I (\(0\)–\(90°\)) | + | + |
| II (\(90\)–\(180°\)) | − | + |
| III (\(180\)–\(270°\)) | − | − |
| IV (\(270\)–\(360°\)) | + | − |
You don’t have to memorise that table if you can picture the point. Left of centre means a negative \(\cos\); below centre means a negative \(\sin\).
Where the wave comes from
Keep spinning past \(360°\) and the point retraces its path, so the values repeat forever. Plot the height against the angle as it goes round and you get the sine curve — smooth, rolling between \(-1\) and 1, repeating every full turn.
That’s the link between circles and waves, and it is why the same two functions describe a pendulum, a sound, a tide, and a rotating motor. Period, amplitude, and phase are all statements about the spinning point: how long a lap takes, how big the circle is, and where the point started.
Radians
Degrees are arbitrary — 360 is a convention. Radians measure angle by arc length around the unit circle instead, so a full turn is \(2\pi\) radians, half a turn is \(\pi\), and a right angle is \(\frac{\pi}{2}\).
The switch looks like extra work and pays for itself later: in calculus, the derivative of \(\sin x\) is \(\cos x\) only when \(x\) is in radians. In degrees an awkward constant shows up and never leaves.
The identity you get for free
The point is on a circle of radius 1, so by Pythagoras \(x^2 + y^2 = 1\), which is:
\[\sin^2 \theta + \cos^2 \theta = 1\]The most used identity in trigonometry is just the circle’s equation with the coordinates renamed.
Common mix-ups
- Swapping the coordinates. Cosine is the horizontal one — it comes first, like \(x\) in \((x, y)\).
- Leaving the calculator in the wrong mode. Degrees and radians give different answers for the same keystrokes.
- Expecting sine to exceed 1. The point never leaves the circle, so both functions stay between \(-1\) and 1.
- Measuring clockwise. Positive angles go anticlockwise from the positive \(x\)-axis.
Check yourself
What are \(\cos 180°\) and \(\sin 180°\)? And why can \(\sin \theta = 1.2\) never happen?
At \(180°\) the point is at \((-1, 0)\), so \(\cos 180° = -1\) and \(\sin 180° = 0\). Sine is a height on a circle of radius 1, so it cannot exceed 1.
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