Signs in the four quadrants
For a point (x, y) on a circle of radius r, we define sin θ = y/r, cos θ = x/r and tan θ = y/x. Because x and y change sign in different quadrants, so do the ratios. The CAST rule records which ratio is positive: quadrant 4 Cosine, quadrant 1 All, quadrant 2 Sine, quadrant 3 Tangent.
Reference angle
The reference angle is the acute angle between the terminal side and the x-axis. The value of any ratio equals its value at the reference angle, with a sign fixed by CAST.
On the unit circle, where the radius r = 1, the coordinates of the point are exactly (cos θ, sin θ). This is a quick way to see why the cosine matches the x-coordinate and the sine matches the y-coordinate, and why each ratio can never be larger than 1 in size. For the special angles 30°, 45° and 60° you should know the exact ratios, so that answers can be left in surd form rather than as rounded decimals.
Key formula /
Reference angle α: Q2 → 180° − θ, Q3 → θ − 180°, Q4 → 360° − θ. Then sin θ = ± sin α, etc., sign from CAST. Also sin²θ + cos²θ = 1.
Worked example
Find cos 210°. The angle lies in quadrant 3, where cosine is negative. Reference angle = 210° − 180° = 30°. So cos 210° = −cos 30° = −√3/2 ≈ −0.866.
If sin θ = 5/13 and θ is obtuse (quadrant 2), then cos θ is negative. Using sin²θ + cos²θ = 1, cos θ = −√(1 − 25/169) = −12/13.
Remember
- CAST: quadrants 4-1-2-3 → Cos, All, Sin, Tan positive.
- Ratio magnitude = value at the reference angle.
- Give the sign before the number.