What is the unit circle?

What is the unit circle?

The unit circle is a circle of radius $1$ centred at the origin. Pick a point on it at angle $\theta$ from the positive x-axis: its x-coordinate is $\cos\theta$ and its y-coordinate is $\sin\theta$. That single picture defines sine and cosine for every angle — not just the ones inside a triangle — and shows why they repeat every full turn.

How do sine and cosine come from the circle?

Draw a right triangle from the point straight down to the x-axis. The hypotenuse is the radius, $1$, so the horizontal leg is $\cos\theta$ and the vertical leg is $\sin\theta$. As the point moves round the circle the same picture works past $90°$, where triangles stop making sense: in the second quadrant $\cos\theta$ is negative (the point is left of the y-axis) while $\sin\theta$ is still positive.

What are tangent, secant and the others?

The other four functions are also lengths in the picture. Tangent is the height where the extended radius meets the vertical line $x = 1$, and it equals $\sin\theta/\cos\theta$. The reciprocals — secant $1/\cos\theta$, cosecant $1/\sin\theta$ and cotangent $\cos\theta/\sin\theta$ — are the matching segments on the other tangent lines. Tangent blows up whenever $\cos\theta = 0$, which is why it has gaps.

Why radians?

A radian measures an angle by the arc it cuts off on the unit circle: an arc of length $\theta$ subtends an angle of $\theta$ radians. A full turn has arc length $2\pi$, so $360° = 2\pi$ rad and $180° = \pi$ rad. Radians make formulas clean — for example the arc length is simply $s = r\theta$ — which is why physics uses them.

How does the circle make a wave?

Let the point travel round the circle and record its height. Against the angle $\theta$ this traces the sine wave. Stretching and shifting it gives the general form:

The same equation describes a mass on a spring, a pendulum for small swings and an alternating current: anything that goes round a circle at a steady rate has a sine wave as its shadow.

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