HKDSE Trigonometry: Sine Rule, Cosine Rule & Bearings
For a triangle with sides a, b, c opposite angles A, B, C, the sine rule is a / sin A = b / sin B = c / sin C and the cosine rule is a² = b² + c² − 2bc cos A. Use the sine rule when you know a side and its opposite angle; use the cosine rule for two sides and the included angle, or for three sides. The area is ½ab sin C.
AB = 8 cm and AC = 5 cm stay fixed while ∠A opens from 20° to 160°. The red side BC follows the cosine rule, and the area ½ × 8 × 5 × sin A is largest at 90°. Try it yourself →
Trigonometry Formulas for HKDSE
Sine rulea / sin A = b / sin B = c / sin C
Cosine rule (side)a² = b² + c² − 2bc cos A
Cosine rule (angle)cos A = (b² + c² − a²) / (2bc)
Area of a triangle½ab sin C; Heron's formula √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2
Identitiessin²θ + cos²θ = 1, tan θ = sin θ / cos θ
BearingsTrue bearing: clockwise from north, e.g. 030°; compass bearing: e.g. N30°E
Sine Rule or Cosine Rule? How to Choose
Two angles and a side (AAS or ASA): sine rule.
Two sides and a non-included angle (SSA): sine rule, then check the ambiguous case.
Two sides and the included angle (SAS): cosine rule for the third side.
Three sides (SSS): cosine rule in the form cos A = (b² + c² − a²) / (2bc).
For problems in solids, such as the angle between a line and a plane, see 3D trigonometry.
True Bearing vs Compass Bearing
A true bearing is measured clockwise from north and written with three digits, such as 045°. A compass bearing starts from north or south and turns towards east or west.
N30°E = 030°
S40°E = 180° − 40° = 140°
S25°W = 180° + 25° = 205°
N70°W = 360° − 70° = 290°
"The bearing of A from B" is measured at B. Bearings in opposite directions differ by 180°.
The Core Concept
How to Solve Any Triangle: When a triangle does not have a 90° angle, we cannot use standard SOH-CAH-TOA. Instead, we must use the Sine Rule or Cosine Rule. Choosing the right tool depends entirely on what information the question gives you.
DSE More about Trigonometry Cheat Sheet — 4 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
Topic
Key Concept & Steps
Exam Shortcuts & Tips
1. Sine Rule
a / sin A = b / sin B = c / sin C
• AAS (2 angles, 1 side) • ASA (2 angles, included side) • SSA (2 sides, non-included angle) Watch out for SSA: This is the only case that can trigger the "ambiguous case" (2 possible triangles).
2. Cosine Rule
• To find a side: a² = b² + c² − 2bc cos A • To find an angle: cos A = (b² + c² − a²) / (2bc)
• SAS (2 sides, included angle) • SSS (All 3 sides known) Use the angle form to check if an angle is obtuse: • If cos A > 0 ⟹ A is acute. • If cos A < 0 ⟹ A is obtuse.
3. Area of Triangle
Area = ½ ab sin C
• SAS (2 sides and their included angle) The angle Cmust be the angle pinched between sides a and b.
4. Bearings
• True Bearing: Measured clockwise from North (000° to 360°). • Compass Bearing: e.g., N30°E or S45°W.
Draw a mini coordinate cross (North-South-East-West) at the "FROM" point.
Worked Examples — More about Trigonometry
Example 1: Cosine rule and the area of a triangle
In △ABC, AB = 8 cm, AC = 5 cm and ∠A = 60°. Find BC and the area of △ABC.
Two sides and the included angle are known, so use the cosine rule: BC² = 8² + 5² − 2(8)(5) cos 60°.
BC² = 64 + 25 − 40 = 49, so BC = 7 cm.
Area = ½ × 8 × 5 × sin 60° = 20 × (√3/2) = 10√3 ≈ 17.3 cm².
Answer:BC = 7 cm; area = 10√3 ≈ 17.3 cm².
Example 2: Bearings with Pythagoras' theorem
A ship sails 30 km from P to Q on a bearing of 060°, then 40 km from Q to R on a bearing of 150°. Find the distance PR and the bearing of R from P.
At Q, the bearing back to P is 060° + 180° = 240°.
So ∠PQR = 240° − 150° = 90°.
PR = √(30² + 40²) = 50 km.
tan ∠QPR = 40/30, so ∠QPR ≈ 53.1°. The bearing of R from P is 060° + 53.1° = 113.1°.
Answer:PR = 50 km; the bearing of R from P is about 113.1° (S66.9°E).
💡 Exam Traps & Secrets
⚠️ The Ambiguous Case (SSA) TrapWhen using the Sine Rule with Side-Side-Angle (SSA), you must check for the ambiguous case if the side opposite to the given angle (a) is shorter than the adjacent side (b) (a < b): 1. Solve for the acute angle: sin B = b sin A / a ⟹ B₁ 2. Find the possible obtuse angle: B₂ = 180° − B₁ 3. Test if B₂ is valid: If A + B₂ < 180°, then two distinct triangles exist, and both answers are correct!
⚠️ Bearings: "Of" vs. "From" TrapOne of the most common mistakes in Paper 1 is drawing the bearing angle from the wrong vertex. "The bearing of A from B is 120°" means you must draw your North arrow and measure the angle starting at point B, not point A.
⚠️ Angles of Elevation & DepressionBoth angles of elevation and depression are measured from the horizontal line of sight. A common mistake is measuring the angle of depression from the vertical wall/cliff instead of the horizontal.
✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Sine Rule SSA — Ambiguous Case)In ΔABC, AB = 7, BC = 9, and ∠BAC = 50°. Find all possible lengths of AC.
(Cosine Rule SSS angle form)In ΔPQR, PQ = 5, QR = 8, and PR = 6. Find ∠PQR.
(Multi-step 2D applications)From a point A on level ground, the angle of elevation of the top of a tower is 35°. Walking 20 m horizontally closer to the tower to point B, the angle of elevation becomes 60°. Find the height of the tower.
More about Trigonometry Exercises: DSE-Style Practice
DSE-style practice sets from the SketchMath question bank, with solutions for every MC and long question.
Sine and Cosine Formulae; 2-D and 3-D Problems · Form 4
Sine Rule: AAS, ASA, or SSA. Cosine Rule: SAS or SSS. For area, use Area = ½ab sinC when you have two sides and the included angle. The key is to scan the given information and match it to the rule that fits.
SSA: two sides and a non-included angle. There can be 0, 1, or 2 valid triangles. Compute sinB from the Sine Rule: if sinB < 1, check both the acute angle B₁ and its supplement B₂ = 180° − B₁. If A + B₂ < 180°, both triangles are valid and you must give both answers.
Once you isolate the correct 2D cross-section in a 3D problem, you apply exactly the same Sine Rule, Cosine Rule, and area formula as in 2D questions. The only hard part is identifying the right triangle from the 3D solid — rotate the model to find it.
The Compulsory Part covers trigonometric ratios and identities, graphs of sin, cos and tan, solving trigonometric equations, the sine and cosine rules, the area formula ½ab sin C, Heron's formula, bearings, angles of elevation and depression, and 3D problems such as the angle between a line and a plane.
A true bearing is measured clockwise from north as a three-digit angle, such as 140°. A compass bearing starts from north or south and turns east or west, such as S40°E. Both describe the same direction: S40°E = 180° − 40° = 140°.