Finding an angle
Type an angle into Go to an angle and the circle jumps to it. It reads degrees by
default (240), multiples of pi (5pi/3, -3pi/4, pi/6), and plain radians when you
add rad (4rad). Angles past a full turn and negative angles are welcome: the readout
names the coterminal angle between 0° and 360° and the reference angle, which are
the two numbers that make an angle like 5pi/3 easy to work with.
Fill in the unit circle turns the same sixteen angles into a blank sheet. Type the
coordinates and the tangent from memory and each box is marked as you go. Answers are
checked by value rather than by spelling, so √3/2, sqrt3/2 and 0.866 all count.
What the unit circle shows
The unit circle is the circle of radius exactly 1 centered on the origin. Its equation is x² + y² = 1. It turns the trigonometric functions into something you can see: for any angle θ measured counter-clockwise from the positive x-axis, the point where the radius meets the circle has coordinates (cos θ, sin θ).
Both halves of that definition are load-bearing. A circle of radius 1 drawn anywhere else on the plane is not the unit circle, because the coordinates of its points are no longer the cosine and sine of the angle. Move the center to (3, 2) and the equation becomes (x − 3)² + (y − 2)² = 1, which describes a perfectly good circle that is no use for trigonometry.
Drag the point above to any angle and watch four things update together:
- The angle, in both degrees and radians.
- cos θ, the point’s horizontal (x) position.
- sin θ, the point’s vertical (y) position.
- tan θ, the ratio sin θ ⁄ cos θ, undefined wherever cos θ = 0 (within one turn, at 90° and 270°).
The labelled unit circle
Every angle worth knowing, on one circle: degrees outside, radians inside.
The angles run counter-clockwise from 0° on the right. That direction is a convention, but it is a universal one, and it is why 90° is at the top and 270° at the bottom rather than the other way round.
Unit circle chart: every angle, coordinate and value
The coordinates are the row to copy. Sine and cosine are just the two halves of the point, listed separately here because that is how questions ask for them.
| Degrees | Radians | Point (cos θ, sin θ) | sin θ | cos θ | tan θ |
|---|---|---|---|---|---|
| 0° | 0 | (1, 0) | 0 | 1 | 0 |
| 30° | π/6 | (√3/2, 1/2) | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | (√2/2, √2/2) | √2/2 | √2/2 | 1 |
| 60° | π/3 | (1/2, √3/2) | √3/2 | 1/2 | √3 |
| 90° | π/2 | (0, 1) | 1 | 0 | undefined |
| 120° | 2π/3 | (-1/2, √3/2) | √3/2 | -1/2 | -√3 |
| 135° | 3π/4 | (-√2/2, √2/2) | √2/2 | -√2/2 | -1 |
| 150° | 5π/6 | (-√3/2, 1/2) | 1/2 | -√3/2 | -√3/3 |
| 180° | π | (-1, 0) | 0 | -1 | 0 |
| 210° | 7π/6 | (-√3/2, -1/2) | -1/2 | -√3/2 | √3/3 |
| 225° | 5π/4 | (-√2/2, -√2/2) | -√2/2 | -√2/2 | 1 |
| 240° | 4π/3 | (-1/2, -√3/2) | -√3/2 | -1/2 | √3 |
| 270° | 3π/2 | (0, -1) | -1 | 0 | undefined |
| 300° | 5π/3 | (1/2, -√3/2) | -√3/2 | 1/2 | -√3 |
| 315° | 7π/4 | (√2/2, -√2/2) | -√2/2 | √2/2 | -1 |
| 330° | 11π/6 | (√3/2, -1/2) | -1/2 | √3/2 | -√3/3 |
Read across 240°: the point is (-1/2, -√3/2), so cos 240° = -1/2 and sin 240° = -√3/2. The x-coordinate is the cosine, and swapping the two is the most common way a filled-in circle goes wrong.
Only five numbers appear in the whole table: 0, 1/2, √2/2, √3/2 and 1. Everything else is one of those with a minus sign in front. That is what makes the circle memorable rather than a list of forty-eight facts.
How to find tan on the unit circle
Tangent is not a coordinate, which is why it feels harder than the other two. It is the ratio of the two coordinates:
tan θ = sin θ ⁄ cos θ = y ⁄ x
Three steps, using 240° as the worked example:
- Find the point. At 240° the circle gives (-1/2, -√3/2).
- Divide y by x. (-√3/2) ÷ (-1/2) = √3. Two negatives make a positive.
- Check the sign against the quadrant. 240° is in quadrant 3, where tangent is positive. It is, so the answer stands: tan 240° = √3.
The shortcut for a special angle is to skip to the reference angle. 240° has a reference angle of 60°, tan 60° = √3, and quadrant 3 makes tangent positive, so tan 240° = √3 without dividing anything.
All tangent values on the unit circle
Read the tangent column of the chart above and it looks like sixteen answers. It is seven, each used twice, plus two angles where tangent has no value at all.
| tan θ | As a decimal | At these angles |
|---|---|---|
| -√3 | -1.732 | 120°, 300° |
| -1 | -1.000 | 135°, 315° |
| -√3/3 | -0.577 | 150°, 330° |
| 0 | 0.000 | 0°, 180° |
| √3/3 | 0.577 | 30°, 210° |
| 1 | 1.000 | 45°, 225° |
| √3 | 1.732 | 60°, 240° |
| undefined | no value | 90°, 270° |
Tangent is undefined at 90° and 270° because the x-coordinate there is 0, and dividing by zero has no answer. Everywhere else on the circle, tangent is one of seven numbers.
Which functions are positive where
The reference angle gives you the number. The quadrant gives you the sign.
| Quadrant | Angles | sin θ | cos θ | tan θ | Positive |
|---|---|---|---|---|---|
| Q1 | 0° to 90° 30°, 45°, 60° | positive | positive | positive | All three |
| Q2 | 90° to 180° 120°, 135°, 150° | positive | negative | negative | Sine only |
| Q3 | 180° to 270° 210°, 225°, 240° | negative | negative | positive | Tangent only |
| Q4 | 270° to 360° 300°, 315°, 330° | negative | positive | negative | Cosine only |
The mnemonic is ASTC, read counter-clockwise from quadrant 1: All, Sine, Tangent, Cosine. It is not arbitrary. Sine is the y-coordinate, so it is positive above the x-axis, which is quadrants 1 and 2. Cosine is the x-coordinate, so it is positive to the right of the y-axis, which is quadrants 1 and 4. Tangent is their ratio, so it is positive wherever the two agree, which is quadrants 1 and 3.
Degrees and radians
Degrees are convenient, but most of mathematics (and all of calculus) uses radians, where a full turn is 2π instead of 360°. One radian is the angle that sweeps out an arc equal to the radius, which makes it about 57.3° and awkward as a unit until you stop writing radians as decimals and start writing them as multiples of π.
One fact converts either way:
180° = π radians
- Degrees to radians: multiply by π⁄180. So 240° × π⁄180 = 4π⁄3.
- Radians to degrees: multiply by 180⁄π. So 5π⁄3 × 180⁄π = 300°.
Turning on snap to common angles in the tool shows the exact radian values (like π⁄6, π⁄4 and π⁄3) instead of decimals, and the radian column of the chart above is the full conversion table for the angles that come up.
The “special” angles worth memorizing
At 30°, 45° and 60° (and their reflections around the circle), sine and cosine take clean exact values built from √2 and √3:
- 30° (π⁄6): sin = 1⁄2, cos = √3⁄2
- 45° (π⁄4): sin = √2⁄2, cos = √2⁄2
- 60° (π⁄3): sin = √3⁄2, cos = 1⁄2
Notice the symmetry as you drag into each quadrant: the values repeat, only the signs change. That single idea, same reference angle with predictable signs, is most of what the unit circle is teaching you, and it is why five memorized values are enough to fill in the whole blank sheet.
How to use this with a class
Project it, drag to an angle, and ask students to predict the sign of sine and cosine before you cross into the next quadrant. Then start the worksheet and work round one quadrant at a time, so the reference-angle pattern is doing the work rather than recall. It is also free to embed on your own site or LMS using the snippet below.