Trigonometry
Right-angled triangle trigonometry: sine, cosine and tangent, exact values for 30°, 45° and 60°, gradients, angles of elevation and depression, and bearings. It is the base of F4 trigonometry and F5 3-D problems.
What you will learn
- 27.1 sin θ, cos θ, tan θ for 0° < θ < 90°.
- 27.2 Properties:
- 27.3 Exact values for 30°, 45°, 60°.
- 27.4 Problems in plane figures.
- 27.5 Gradients, angles of elevation and depression, bearings (both 010° and N10°E forms).
You need first: Geometry and data basics, Similar triangles, Surds
Key terms
| English | 中文 |
|---|---|
| sine, cosine, tangent | 正弦，餘弦，正切 |
| trigonometric ratios | 三角比 |
| opposite side, adjacent side, hypotenuse | 對邊，鄰邊，斜邊 |
| angle of elevation / depression | 仰角／俯角 |
| bearing | 方位角 |
| true bearing, compass bearing | 真方位，羅盤方位 |
| gradient, inclination | 傾斜度，傾角 |
Standard steps
Choose the ratio
| Ratio | Uses | Memory |
|---|---|---|
| sin θ = opposite ÷ hypotenuse | SOH | |
| cos θ = adjacent ÷ hypotenuse | CAH | |
| tan θ = opposite ÷ adjacent | TOA |
- Mark the angle θ.
- Label the sides from θ's point of view: opposite (across from θ), adjacent (next to θ, not the hypotenuse), hypotenuse (across from the right angle).
- Circle the side you know and the side you want. Pick the ratio that uses both.
- Write the equation and solve. To find an angle, use sin⁻¹, cos⁻¹ or tan⁻¹.
Why it works: every right triangle with the same angle θ is similar, so the ratio of its sides is always the same number.
Exact values
| θ | 30° | 45° | 60° |
|---|---|---|---|
| sin θ | 1/2 | √2/2 | √3/2 |
| cos θ | √3/2 | √2/2 | 1/2 |
| tan θ | √3/3 | 1 | √3 |
Build them from half an equilateral triangle (sides 1, √3, 2) and half a square (sides 1, 1, √2), rather than memorising.
Elevation, depression, bearings
- Draw a sketch. Mark the horizontal line first.
- Elevation is measured up from the horizontal; depression down from the horizontal.
- Bearings: true bearings are measured clockwise from north, with three digits (053°). Compass bearings start from N or S and turn towards E or W (N53°E).
- Gradient = vertical rise ÷ horizontal run = tan(inclination). "1 : 4" means rise 1, run 4.
Worked examples
tan 32° = h / 50
h = 50 tan 32°
≈ 31.2 m (3 sig. fig.)tan θ = 1/4, θ ≈ 14.0°.
Distance = √(3² + 4²) = 5 km. tan θ = 4/3, θ ≈ 53.1°, so the bearing is 053.1° (N53.1°E).
= sin²θ / (sin θ cos θ) sin²θ + cos²θ = 1
= sin θ / cos θ
= tan θ= (√3/2)(√3/2) − 1 = 3/4 − 1 = −1/4.
Common mistakes
- Labelling "adjacent" and "opposite" from the wrong angle.
- Calculator in radians (RAD) instead of degrees (DEG).
- Angle of depression drawn from the vertical.
- Bearings measured anticlockwise, or written without three digits.
- sin²θ typed as sin(θ²).
- Rounding the angle in the middle and then using it again.
How the DSE tests it
- Paper 2: simplify expressions with sin(90° − θ) and sin²θ + cos²θ = 1; elevation, depression and bearing items.
- Paper 1: right-triangle work inside longer geometry and 3-D questions.
- Foundation students who skip the NF sine and cosine formulas still use this unit everywhere.
Questions you might not ask in class
Where do sin, cos and tan come from? Are they just buttons?
They are fixed ratios of side lengths for each angle. Ancient astronomers made tables of them; the calculator just looks them up.
How do I know whether to use sin or sin⁻¹?
If you're finding a side, use sin. If you're finding an angle, use sin⁻¹.
Why does depression equal elevation?
They are alternate angles between two horizontal (parallel) lines.