Why we need ratios beyond 90°
In a right triangle, the other two angles are less than 90°. So sin, cos and tan were only defined for acute angles. But real triangles can have an obtuse angle (more than 90°). To measure them, we need sin, cos and tan for angles up to 180°.
The trick: use a circle of radius 1 with its centre at the origin. This is the unit circle. For an acute angle θ, a point P on the circle makes a small right triangle with the x-axis. Its height is sin θ and its base is cos θ. We keep exactly this idea and let θ grow past 90°.
Defining sin, cos and tan with the unit circle
Measure the angle θ from the positive x-axis, turning anticlockwise. Let P = (x, y) be the point on the circle.
- cos θ = x (sideways distance, can be negative)
- sin θ = y (height, positive for 0° to 180°)
- tan θ = y ÷ x = sin θ ÷ cos θ (when x is not 0)
For acute angles this gives the same answers as the right triangle. For 90° the point is at (0, 1): cos 90° = 0, sin 90° = 1, and tan 90° is not defined because we cannot divide by 0.
Signs and the mirror rule: 180° − θ
For an obtuse angle, P is on the left, so x is negative.
- sin is positive (0° to 180°)
- cos is negative (90° to 180°)
- tan is negative (90° to 180°), because tan = sin ÷ cos is positive ÷ negative
The point at 180° − θ is the mirror of the point at θ across the y-axis. So:
sin(180° − θ) = sin θ, cos(180° − θ) = −cos θ, tan(180° − θ) = −tan θ.
Use it to find any obtuse value from an acute one: sin 120° = sin 60° = √3/2; cos 120° = −cos 60° = −1/2; tan 120° = −tan 60° = −√3.
Values at 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°
| θ | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° |
|---|---|---|---|---|---|---|---|---|---|
| sin | 0 | 1/2 | √2/2 | √3/2 | 1 | √3/2 | √2/2 | 1/2 | 0 |
| cos | 1 | √3/2 | √2/2 | 1/2 | 0 | −1/2 | −√2/2 | −√3/2 | −1 |
| tan | 0 | √3/3 | 1 | √3 | – | −√3 | −1 | −√3/3 | 0 |
Two useful facts: sin 0° = sin 180° = 0, and cos 180° = −1.
Try it
Open a door or a laptop lid. Stand it at 60°, then at 120°. Look at the tip from above: the height from the table is the same, but the tip is on the other side of the hinge. Now in the 3D, slide the angle to 60° and then 120° and compare sin and cos.
Key formulas and definitions
- cos θ = x, sin θ = y, tan θ = y/x (point on unit circle)
- sin(180° − θ) = sin θ
- cos(180° − θ) = −cos θ
- tan(180° − θ) = −tan θ
- tan θ = sin θ / cos θ (not defined when cos θ = 0, i.e. θ = 90°)
- sin²θ + cos²θ = 1 for every θ
Worked examples
1. Find sin 135°.
sin 135° = sin(180° − 45°) = sin 45° = √2/2 ≈ 0.71.
2. Find cos 150°.
cos 150° = −cos(180° − 150°) = −cos 30° = −√3/2 ≈ −0.87.
3. Find tan 120°.
tan 120° = −tan 60° = −√3 ≈ −1.73. (Check: sin 120° ÷ cos 120° = (√3/2) ÷ (−1/2) = −√3.)
4. θ is between 0° and 180° and sin θ = √3/2. Find all possible θ.
sin 60° = √3/2. The mirror angle 180° − 60° = 120° has the same sin. So θ = 60° or 120°.
5. sin θ = 3/5 and θ is obtuse. Find cos θ and tan θ.
cos²θ = 1 − 9/25 = 16/25, so cos θ = ±4/5. For an obtuse angle cos is negative: cos θ = −4/5. tan θ = (3/5) ÷ (−4/5) = −3/4.
6. A triangle has two sides 6 cm and 8 cm and the angle between them is 150°. Find its area using ½ab sin C.
Area = ½ × 6 × 8 × sin 150° = 24 × ½ = 12 cm².
7. tan θ = −1 for 0° < θ < 180°. Find θ.
tan is negative, so θ is obtuse. tan 45° = 1, so θ = 180° − 45° = 135°.
Common mistakes
- Writing cos 120° = cos 60°. The sign flips: cos 120° = −cos 60° = −1/2.
- Thinking sin becomes negative for obtuse angles. Between 0° and 180° sin is never negative.
- Saying tan 90° = 0 or ∞ as a number. It is simply not defined, because cos 90° = 0.
- Measuring the obtuse angle from the negative x-axis without the 180° − θ step. Always start from the positive x-axis.