All four quadrants
What you'll learn
Use reference angles and the ASTC rule to find the sign of sin, cos, and tan in any quadrant.
The unit circle point is (cos θ, sin θ). As θ moves past 90°, the point moves into new quadrants — and the x and y coordinates change sign. That's all that changes: the magnitudes follow the same triangles, the signs depend on the quadrant.
Signs by quadrant
The mnemonic ASTC — "All Students Take Calculus" — tells you which ratios are positive:
| Quadrant | Angles | Positive ratios |
|---|---|---|
| I (top-right) | 0° – 90° | All (sin +, cos +, tan +) |
| II (top-left) | 90° – 180° | Sin only |
| III (bottom-left) | 180° – 270° | Tan only (both sin and cos negative, so tan = +) |
| IV (bottom-right) | 270° – 360° | Cos only |
Reference angles
For any angle θ, the reference angle is the acute angle between the radius and the nearest x-axis. It's always between 0° and 90°.
The trig values of θ have the same magnitude as the trig values of the reference angle — only the signs differ (from the quadrant table).
| Quadrant | Reference angle |
|---|---|
| I | θ |
| II | 180° − θ |
| III | θ − 180° |
| IV | 360° − θ |
Example: sin 150°. Q II → reference angle = 180° − 150° = 30°. sin 30° = 0.5, and sin is positive in Q II, so sin 150° = +0.5.
Angles beyond 360°
Going past 360° just wraps around: 390° is the same point as 30°, so the trig values are identical. Negative angles go clockwise: −90° lands at the same point as 270°.
Check your understanding
Question 1 of 2
In which quadrant are both sin θ and cos θ negative?