The unit circle and radians

1. The problem with triangles

In a right triangle, the two other angles are always less than 90°. So SOH CAH TOA only tells you sin, cos and tan for angles between 0° and 90°.

But angles go further. A wheel turns 200°, a door swings 120°, a clock hand goes all the way round. We want sin and cos for any angle. The unit circle is how we get them, and for angles under 90° it gives exactly the same answers as the triangle.

2. Radians: measuring an angle by distance

Degrees split a full turn into 360 pieces. That number is a choice someone made long ago. Radians measure an angle by something natural: how far you travel around a circle of radius 1.

Drag the slider. The orange arc is the distance traveled around the circle. Its length is the angle in radians.

Going all the way round a circle of radius 1 is a distance of 2π (that's the circumference, 2πr with r = 1). So:

To convert: degrees → radians, multiply by π/180. Radians → degrees, multiply by 180/π (or just replace π with 180°). For example, 5π/6 = 5 × 180°/6 = 150°.

Radians are usually written as a fraction of π, like π/6, because the special angles come out neat that way. π/6 is exactly 30°, while "0.5236" hides that.

3. The unit circle: cos is x, sin is y

Draw a circle of radius 1 centered at the origin. Start at the point (1, 0) and turn counterclockwise by an angle θ. Where you land is a point (x, y). Drop a line from it to the x-axis and you get a right triangle with hypotenuse 1.

SOH CAH TOA with a hypotenuse of 1 says sin θ = opposite / 1 = y and cos θ = adjacent / 1 = x. That gives the most important fact on this page:

The point at angle θ is (cos θ, sin θ).

That works for any angle, not just ones under 90°. Past 90° there's no triangle to start from, but the point still has an x and a y. So we define cos θ as x and sin θ as y. Try it:

Notice: in the left half x is negative, so cos is negative there. In the bottom half y is negative, so sin is negative there.

4. The special angles

Most angles give messy decimals. But 30°, 45° and 60° come from two famous triangles, and their values are worth knowing by heart:

A memory trick: going down the sin column, the values are √0/2, √1/2, √2/2, √3/2, √4/2, which simplify to 0, 1/2, √2/2, √3/2, 1. The cos column is the same list upside down.

Where they come from: cut a square along its diagonal and you get a 45-45-90 triangle; with hypotenuse 1, both legs are √2/2. Cut an equilateral triangle in half and you get a 30-60-90 triangle; with hypotenuse 1, the short side is 1/2 and the long side is √3/2.

5. Any angle: reference angle plus sign

Every angle's point is a mirror image of a point in the first quadrant. So for any angle you need two things:

1. The size comes from the reference angle: the acute angle between θ's line and the x-axis (the nearest side of it).
2. The sign comes from the quadrant: is x left or right, is y up or down?

The "positive" column is often remembered as All Students Take Calculus, going counterclockwise from quadrant I: All, Sin, Tan, Cos.

Example: sin(210°). 210° is in quadrant III, so the reference angle is 210° − 180° = 30°, and sin(30°) = 1/2. In quadrant III only tan is positive, so sin(210°) = −1/2.

Going past a full turn, or backwards: adding or subtracting 360° lands on the same point (these are called coterminal angles). So 390° acts like 30°, and −150° acts like 210°. Negative angles just turn clockwise.

6. Worked examples

Each one is freshly generated. Press Another example as often as you like.