The unit circle is a circle of radius 1 centered at the origin. For any angle θ measured from the positive x-axis, the point where the angle’s ray hits the circle is (cos θ, sin θ). That one fact lets sine and cosine work for any angle, including angles bigger than 90° and negative angles.
You don’t need to memorize the whole circle. Learn three points in the first quadrant and two rules, and you can rebuild the rest.
Degrees and radians
A full turn is 360°, or 2π radians. So 180° = π radians.
- Degrees to radians: multiply by π/180.
- Radians to degrees: multiply by 180/π.
Example Convert 60° to radians and 3π/4 to degrees
60 × π/180 = π/3. And 3π/4 × 180/π = 135°.
Radians measure an angle by the arc length it cuts on a unit circle. That’s why calculus uses them: the formulas come out cleaner.
The first quadrant
| Degrees | Radians | Point (cos θ, sin θ) |
|---|---|---|
| 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) |
A pattern helps: the sines of 0°, 30°, 45°, 60°, 90° are √0/2, √1/2, √2/2, √3/2, √4/2. Cosine runs the same list backwards.
Every other angle: reference angles
The reference angle is the acute angle between the terminal side and the x-axis. An angle and its reference angle have the same sine and cosine values apart from the sign.
| Quadrant | Reference angle (degrees) |
|---|---|
| I | θ |
| II | 180° − θ |
| III | θ − 180° |
| IV | 360° − θ |
Then attach the signs. In each quadrant, the signs are the signs of x and y there:
| Quadrant | cos (x) | sin (y) | tan |
|---|---|---|---|
| I | + | + | + |
| II | − | + | − |
| III | − | − | + |
| IV | + | − | − |
Some students remember the positive functions with “All Students Take Calculus”: All in I, Sine in II, Tangent in III, Cosine in IV.
Example Find the point for 150°
- Quadrant II, reference angle 180° − 150° = 30°.
- The 30° point is (√3/2, 1/2).
- In Quadrant II, x is negative and y is positive: (−√3/2, 1/2).
Example Find the point for 225°
Quadrant III, reference angle 45°. Both coordinates are negative: (−√2/2, −√2/2).
Example Find sin 300°
Quadrant IV, reference angle 60°, and sine is negative there: sin 300° = −√3/2.
Tangent
tan θ = sin θ / cos θ, which is y/x on the circle. It is the slope of the angle’s ray.
Example Find tan 120°
The point is (−1/2, √3/2), so tan 120° = (√3/2) / (−1/2) = −√3.
Tangent is undefined at 90° and 270°, where cos θ = 0.
Coterminal and negative angles
Angles that end in the same place are coterminal. Add or subtract 360° (or 2π) to find one. 405° ends where 45° does, and −π/6 ends where 11π/6 does.
A negative angle is measured clockwise. −45° lands in Quadrant IV, at (√2/2, −√2/2).
The identity that comes free
Every point on the unit circle is 1 unit from the origin. The Pythagorean theorem on the point (cos θ, sin θ) gives:
sin²θ + cos²θ = 1
Example If sin θ = 3/5 and θ is in Quadrant II, find cos θ
- cos²θ = 1 − 9/25 = 16/25, so cos θ = ±4/5.
- Cosine is negative in Quadrant II, so cos θ = −4/5.
How to learn it for good
- Memorize only the first-quadrant table.
- Practice finding reference angles until it’s automatic.
- Practice the signs by thinking “x is cosine, y is sine” instead of memorizing a chart.
- Draw the circle from memory once a day for a week. Spaced practice like this is what makes it stick; see Spaced Repetition.
Try these
Convert 210° to radians.
Show answer
7π/6.
Convert 5π/3 to degrees.
Show answer
300°.
Find sin 135°.
Show answer
√2/2. Reference angle 45°, and sine is positive in Quadrant II.
Find cos(4π/3).
Show answer
−1/2. 4π/3 is 240°, reference angle 60°, and cosine is negative in Quadrant III.
Find tan 330°.
Show answer
−√3/3. The point is (√3/2, −1/2), so y/x = −1/√3.
Find a positive angle coterminal with −45°.
Show answer
315°.
Practice
Questions students ask
Why is the radius 1?
With a radius of 1, the hypotenuse of every triangle in the circle is 1, so cos θ and sin θ are exactly the x and y coordinates with no dividing needed. A circle of radius r has the point (r cos θ, r sin θ).
Do I need radians or degrees?
Both. Geometry and many test questions use degrees. Precalculus and calculus use radians almost all the time.
What comes before this?
Right-triangle trig. If SOH CAH TOA isn’t solid yet, start with Right-Triangle Trigonometry.