What is the unit circle?

The unit circle is a circle of radius 1 centred at the origin. For a point on it at angle θ (measured anticlockwise from the positive x-axis), the x-coordinate is cos θ and the y-coordinate is sin θ. It is the single definition behind all of trigonometry.

How does the unit circle relate to the sine graph?

As the point travels around the circle, its height above the centre is sin θ. Plotting that height against the angle θ traces out the sine wave. The cosine graph is the same idea using the horizontal position.

Why do sine and cosine repeat every 360° (2π)?

Because after a full turn around the circle the point returns to exactly where it started, so its coordinates repeat. That is why sine and cosine are periodic with period 360° or 2π radians.

A radian is the angle subtended when the arc length equals the radius. A full circle is 2π radians ≈ 6.28, so 2π rad = 360°, π rad = 180°. Radians make calculus of trig functions clean, which is why they are standard beyond GCSE.

The Unit Circle Explained (Interactive) — Where Sine & Cosine Come From

Sine and cosine can feel like arbitrary buttons on a calculator. They aren’t. Both come from one picture: a point moving around the unit circle. Sine is its height; cosine is its width. Once you’ve watched the sine wave get traced out by a spinning point, trig graphs stop being something to memorise.

One picture for all of trig

The unit circle is the circle of radius $1$ centred at the origin. Put a point on it and let $\theta$ be the angle from the positive $x$ -axis, measured anticlockwise. Then, by definition, the point’s coordinates are

(\cos\theta,\ \sin\theta).

That’s the whole idea. Cosine is the horizontal coordinate, sine is the vertical coordinate. Everything else in trigonometry — the graphs, the identities, the exact values — follows from this one definition.

Sine is how high the point is. Cosine is how far across it is.

Trace the wave

Press play (or drag the point) and watch the connector on the right. As the point rotates, its height — $\sin\theta$ — is plotted against the angle, drawing the sine wave in real time. Switch to $\cos\theta$ to trace the cosine wave from the point’s width instead.

InteractiveThe unit circle → the sine wave

Loading interactive…

The height of the rotating point is sin θ. Plotting height against angle traces the sine curve.

Text description ↓

An interactive unit circle with a point at angle θ. The point’s vertical position equals sin θ and its horizontal position equals cos θ. As θ increases, the height is plotted against the angle to trace the sine (or cosine) wave. Key values:

θ	0	π/2 (90°)	π (180°)	3π/2 (270°)
sin θ	0	1	0	−1
cos θ	1	0	−1	0

Sine, cosine and tangent

The circle definition agrees with the right-angled-triangle one you met first (SOH-CAH-TOA). Drop a vertical from the point to the $x$ -axis and you get a right triangle with hypotenuse $1$ : the opposite side has length $\sin\theta$ and the adjacent side $\cos\theta$ . The circle version simply extends this beyond $90^\circ$ , giving meaning to the sine of obtuse and negative angles.

Tangent is their ratio — the slope of the radius:

\tan\theta = \frac{\sin\theta}{\cos\theta}.

Why radians?

A radian measures an angle by arc length: one radian is the angle whose arc equals the radius. Since the full circumference is $2\pi r$ , a full turn is $2\pi$ radians. So

2\pi\text{ rad} = 360^\circ,\qquad \pi\text{ rad}=180^\circ,\qquad \tfrac{\pi}{2}\text{ rad}=90^\circ.

Radians aren’t just a different unit for its own sake — they make calculus of trig functions clean (for example $\frac{d}{dx}\sin x = \cos x$ only in radians). That’s why they become standard from A-Level onward.

Reading the graphs

Everything about the sine and cosine graphs is now explainable from the circle:

They oscillate between $-1$ and $1$ because the point never leaves a radius-1 circle.
They repeat every $360^\circ$ ( $2\pi$ ) because after a full turn the point is back where it started.
Cosine is sine shifted left by $90^\circ$ : $\cos\theta = \sin(\theta+90^\circ)$ , because width leads height by a quarter-turn.

Common mistakes

1Worked example — a value beyond 90°

Find $\cos 120^\circ$ . At $120^\circ$ the point is in the second quadrant, where $x$ is negative. It has the same width as $60^\circ$ but on the left, so $\cos 120^\circ = -\cos 60^\circ = -\tfrac{1}{2}$ .

Your turn

Without a calculator, what is $\sin 180^\circ$ ? Use the circle.

Show the answer ↓

At $180^\circ$ the point sits at $(-1, 0)$ — on the axis, zero height. So $\sin 180^\circ = 0$ .

One picture for all of trig

(\cos\theta,\ \sin\theta).

Sine is how high the point is. Cosine is how far across it is.

Trace the wave

InteractiveThe unit circle → the sine wave

Loading interactive…

The height of the rotating point is sin θ. Plotting height against angle traces the sine curve.

Text description ↓

θ	0	π/2 (90°)	π (180°)	3π/2 (270°)
sin θ	0	1	0	−1
cos θ	1	0	−1	0

Sine, cosine and tangent

Tangent is their ratio — the slope of the radius:

\tan\theta = \frac{\sin\theta}{\cos\theta}.

Why radians?

A radian measures an angle by arc length: one radian is the angle whose arc equals the radius. Since the full circumference is $2\pi r$ , a full turn is $2\pi$ radians. So

2\pi\text{ rad} = 360^\circ,\qquad \pi\text{ rad}=180^\circ,\qquad \tfrac{\pi}{2}\text{ rad}=90^\circ.

Reading the graphs

Everything about the sine and cosine graphs is now explainable from the circle:

They oscillate between $-1$ and $1$ because the point never leaves a radius-1 circle.
They repeat every $360^\circ$ ( $2\pi$ ) because after a full turn the point is back where it started.
Cosine is sine shifted left by $90^\circ$ : $\cos\theta = \sin(\theta+90^\circ)$ , because width leads height by a quarter-turn.

Common mistakes

1Worked example — a value beyond 90°

Your turn

Without a calculator, what is $\sin 180^\circ$ ? Use the circle.

Show the answer ↓

At $180^\circ$ the point sits at $(-1, 0)$ — on the axis, zero height. So $\sin 180^\circ = 0$ .

Sine & Cosine from the Unit Circle

One picture for all of trig

Trace the wave

Sine, cosine and tangent

Why radians?

Reading the graphs

Common mistakes

Frequently asked questions

Keep exploring

Want a tutor to walk you through it?

Sine & Cosine from the Unit Circle

One picture for all of trig

Trace the wave

Sine, cosine and tangent

Why radians?

Reading the graphs

Common mistakes

Frequently asked questions

Keep exploring

Want a tutor to walk you through it?