When do I use Sine Rule vs Cosine Rule?

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.

What is the ambiguous case and how do I check for it?

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.

How does 2D trigonometry link to 3D geometry in DSE?

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.

More about Trigonometry

Q: How does 2D trigonometry link to 3D geometry in DSE?

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.

Master the Sine and Cosine Rules, solve non-right-angled triangles, and conquer the notorious "ambiguous case" without getting lost in the formulas.

1. 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.

2. 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 `C` must 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.

Interactive More about Trigonometry graph — click to explore

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💡 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.

Frequently Asked Questions

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