Reading the Unit Circle
The unit circle is the single most useful picture in all of trigonometry. It is nothing more than a circle of radius 1 centred on the origin of the coordinate plane, yet it manages to turn every angle into a pair of coordinates and every trig value into something you can literally point at. Spend an afternoon reading it fluently and a huge amount of the subject clicks into place.
Why a point is (cos θ, sin θ)
Start at the point (1, 0) on the far right and rotate counterclockwise through an angle θ. The radius to your new point acts as the hypotenuse of a little right triangle dropped down to the x-axis. Because that hypotenuse has length 1, the horizontal leg is cos θ = adjacent / 1, which is just the x-coordinate, and the vertical leg is sin θ = opposite / 1, which is the y-coordinate. So every point on the circle is exactly (cos θ, sin θ). The circle is a lookup table for both functions at once.
The key angles
Five angles in the first quadrant do most of the heavy lifting. Here they are in degrees, in radians, and as coordinates:
| Degrees | Radians | (cos, sin) |
|---|---|---|
| 0° | 0 | (1, 0) |
| 30° | π/6 | (√3/2, 1/2) |
| 45° | π/4 | (√2/2, √2/2) |
| 60° | π/3 | (1/2, √3/2) |
| 90° | π/2 | (0, 1) |
Where the exact values come from
These are not random decimals to memorise; they fall out of two simple triangles. For 45°, picture a right triangle with two equal legs. Its hypotenuse is √2 times a leg, so both sine and cosine equal 1/√2 = √2/2 ≈ 0.7071 — equal, as symmetry demands. For 30° and 60°, slice an equilateral triangle down the middle. The half you keep has a hypotenuse of 1, a short side of 1/2, and a long side of √3/2. Reading the ratios off that half-triangle gives sin 30° = 1/2 and sin 60° = √3/2, with cosine swapping the two. That is the whole story behind the table.
Signs in the four quadrants
Because cosine is the x-coordinate and sine is the y-coordinate, their signs simply follow the signs of x and y in each quadrant:
- Quadrant I (0° to 90°): both positive.
- Quadrant II (90° to 180°): cos θ < 0, sin θ > 0.
- Quadrant III (180° to 270°): both negative.
- Quadrant IV (270° to 360°): cos θ > 0, sin θ < 0.
The old mnemonic "All Students Take Calculus" names which function stays positive in each quadrant in turn: All, Sine, Tangent, Cosine.
How tangent fits in
Tangent is the ratio of the other two: tan θ = sin θ / cos θ = y / x. Read geometrically, that is the slope of the radius line drawn out to the point. It is positive where x and y share a sign (quadrants I and III) and negative where they differ (II and IV). At 90° and 270° the x-coordinate is 0, so the division blows up and tangent is undefined — the radius there is perfectly vertical and has no finite slope.
Reaching the rest of the circle
You only ever memorise the first quadrant; the other three come free through reference angles. A reference angle is the acute gap between your radius and the x-axis. Take 150°: it sits in quadrant II, and its reference angle is 180° − 150° = 30°. So its coordinates match 30° in size, (√3/2, 1/2), and you only adjust the signs for quadrant II, where cosine turns negative. That gives (−√3/2, 1/2), so cos 150° = −√3/2 and sin 150° = 1/2. Every angle in the full 360°, and beyond, reduces to a first-quadrant value dressed with the correct sign.
The circle rewards slow, repeated reading. Explore every angle on it with the calculators on SinCosTan.