The circle that defines trig
The unit circle has radius 1 and centre (0, 0). An angle θ in standard position starts on the positive x-axis and rotates counterclockwise (negative angles rotate clockwise). The terminal arm meets the circle at a point:
P(θ) = (cos θ, sin θ)
and tan θ = sin θ / cos θ = y/x.
So on the unit circle, cosine is the x-coordinate and sine is the y-coordinate.
Key points on the axes
| θ | Point | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0° (0) | (1, 0) | 1 | 0 | 0 |
| 90° (π/2) | (0, 1) | 0 | 1 | undefined |
| 180° (π) | (−1, 0) | −1 | 0 | 0 |
| 270° (3π/2) | (0, −1) | 0 | −1 | undefined |
Special angles in quadrant I
| θ | cos θ | sin θ | tan θ |
|---|---|---|---|
| 30° (π/6) | √3/2 | 1/2 | √3/3 |
| 45° (π/4) | √2/2 | √2/2 | 1 |
| 60° (π/3) | 1/2 | √3/2 | √3 |
These come from the 30–60–90 and 45–45–90 triangles scaled to hypotenuse 1.
Reference angles
For any angle, the reference angle is the acute angle to the x-axis. The trig values have the same size as those of the reference angle; only the signs change.
| Quadrant | Reference angle |
|---|---|
| I | θ |
| II | 180° − θ |
| III | θ − 180° |
| IV | 360° − θ |
Signs: the CAST rule
Starting in quadrant IV and moving counterclockwise:
| Quadrant | Positive |
|---|---|
| I | All |
| II | Sine |
| III | Tangent |
| IV | Cosine |
That’s the CAST rule.
Example
Find the exact value of cos 150°.
- Quadrant II; reference angle 30°
- cos 30° = √3/2; cosine is negative in II
- cos 150° = −√3/2
Example: sin(5π/4)
- Quadrant III; reference angle π/4
- sin is negative → −√2/2
Coterminal angles
Angles that share a terminal arm differ by multiples of 360° (2π): 30°, 390° and −330° all have the same trig values.
The Pythagorean identity
Since (cos θ, sin θ) is on a circle of radius 1:
cos²θ + sin²θ = 1
Worked examples
Example 1: sin 240°? Quadrant III, ref 60° → −√3/2.
Example 2: tan 315°? Quadrant IV, ref 45° → −1.
Example 3: cos(2π/3)? Quadrant II, ref π/3 → −1/2.
Common mistakes
- Mixing up which coordinate is sine (y) and which is cosine (x).
- Wrong signs in different quadrants.
- Forgetting coterminal angles.
Try it yourself
- Coordinates of P(60°) on the unit circle?
- Exact value of sin 135°.
- Exact value of cos 300°.
- tan(7π/6)?
- Find two angles between 0° and 360° with sin θ = −1/2.
Answers: 1) (1/2, √3/2) 2) √2/2 3) 1/2 4) √3/3 5) 210° and 330°
Quick recap
- On the unit circle, P(θ) = (cos θ, sin θ).
- Use reference angles, special angles and the CAST rule for exact values.
- cos²θ + sin²θ = 1.