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Sine/Cosine Rules & Bearings

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

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First Principles

Objective

Apply the sine and cosine rules and solve bearings problems in non-right-angled triangles.

Concrete scenario: A coastguard station at Mombasa port sees Ship A on a bearing of 030° (4 km away) and Ship B on a bearing of 110° (6 km away). How far apart are the two ships? You cannot drop a perpendicular neatly here — triangle OAB is not right-angled. You need general non-right triangle rules.

Geometric insight: Every non-right-angled triangle can be split into two right-angled triangles by dropping an altitude. When you apply Pythagoras to both halves, the altitude cancels out to yield the Cosine Rule: \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Equating the altitude in terms of sines yields the Sine Rule: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\).

Bearings & Triangles: A bearing is a clockwise angle measured from true North (000° to 360°). The interior angle formed between two routes from the same point is the absolute difference between their bearings (or \(360^{\circ}\) minus that difference if greater than \(180^{\circ}\)).

Interactive Bearings & Cosine Rule Explorer

NORTHShip A (4km)Ship B (6km)Port (O)030°110°

Interior Angle \(\Delta\): \(110^{\circ} - 30^{\circ} = 80^{\circ}\)

Distance AB (Cosine Rule):

\(d = \sqrt{4^2 + 6^2 - 2(4)(6)\cos(80^{\circ})} \approx 6.45\,\text{km}\)

Key Formulas

\[\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\] — Sine Rule: Use when matching pairs of side and opposite angle are known (e.g., AAS, ASA, or SSA).
\[c^{2}=a^{2}+b^{2}-2ab\cos C\] — Cosine Rule (Missing Side): Use when two sides and the included angle (SAS) are given.
\[\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}\] — Cosine Rule (Missing Angle): Use when all three sides (SSS) are known.
\[\text{Interior Angle}=|\text{Bearing}_1-\text{Bearing}_2|\] — If difference \(> 180^{\circ}\), use \(360^{\circ} - \text{difference}\).

Worked Examples

Example 1 (Easy) — Cosine Rule for a missing side:

Problem: In triangle ABC, \(a=8\,\text{km}\), \(b=6\,\text{km}\), and \(\angle C=60^{\circ}\). Find side \(c\).

  1. Formula: \[c^{2}=a^{2}+b^{2}-2ab\cos C\]
  2. Substitute values: \[c^{2}=8^{2}+6^{2}-2(8)(6)\cos 60^{\circ}\]
  3. Calculate: \(\cos 60^{\circ}=0.5\), so \[c^{2}=64+36-48=52\]
  4. Solve for \(c\): \[c=\sqrt{52}\approx 7.21\,\text{km}\]
Example 2 (Medium) — Bearings and distance between two ships:

Problem: From Mombasa lighthouse O, Ship A is on bearing 045° at 120 nautical miles. Ship B is on bearing 125° at 80 nm. Find distance AB.

  1. Interior angle at O: \(\Delta = 125^{\circ} - 45^{\circ} = 80^{\circ}\).
  2. Apply Cosine Rule: \[D^{2}=120^{2}+80^{2}-2(120)(80)\cos 80^{\circ}\]
  3. Evaluate: \[D^{2}=14400+6400-19200(0.1736)=20800-3333.12=17466.88\]
  4. Square root: \[D=\sqrt{17466.88}\approx 132.2\,\text{nm}\]
Example 3 (Hard) — Multi-step bearings and Sine Rule:

Problem: A vessel sails 30 km from Port P on bearing 040° to point Q, then 40 km on bearing 130° to point R. Find the total direct bearing from Port P to point R.

  1. Recognize the right angle at Q: The interior turn angle at Q is \(180^{\circ} - (130^{\circ} - 40^{\circ}) = 90^{\circ}\).
  2. Calculate triangle angle \(\angle QPR\): \[\tan(\angle QPR) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{40}{30} = 1.3333 \implies \angle QPR = \tan^{-1}(1.3333) \approx 53.13^{\circ}\]
  3. Add initial bearing to angle: \[\text{Bearing of R from P} = 40^{\circ} + 53.13^{\circ} = 93.13^{\circ} \approx 93^{\circ}\]

Common Mistakes

Mistake Assuming bearings run anti-clockwise like coordinate geometry angles.
Correction Bearings always measure clockwise starting from true North (000°).
Why it feels right Standard Cartesian trigonometry measures angles counter-clockwise from the positive x-axis.
Mistake Using an interior angle greater than \(180^{\circ}\) in the Cosine Rule when calculating bearing differences.
Correction A triangle interior angle cannot exceed \(180^{\circ}\). If \(|\text{Bearing}_1 - \text{Bearing}_2| > 180^{\circ}\), subtract the difference from \(360^{\circ}\).
Why it feels right Plugging raw subtraction results straight into the formula without checking physical geometry.

Real World

Maritime Navigation in Mombasa: Coastguards track cargo vessels outside Kilindini Harbour, calculating relative distance and avoiding collisions using bearings and the Cosine Rule.
Aviation Tracking: Air traffic controllers in Nairobi calculate flight paths when aircraft vector around thunderstorms using radial bearings from VOR radio beacons.
Land Surveying: Surveyors measuring non-accessible terrain across Rift Valley cliffs measure two accessible baselines and an angle to determine gorge widths.

Practice

A triangle has sides a = 13 cm, b = 14 cm, and c = 15 cm. Using the Cosine Rule, find the measure of angle C opposite side c to the nearest whole degree. (Type only the number, e.g., 67)
Review the concepts above.
In triangle PQR, side PQ = 8 cm, side PR = 6 cm, and the included angle \(\angle QPR = 40^{\circ}\). Using the Cosine Rule, calculate the length of side QR to two decimal places. (Type only the number, e.g., 5.14)
Review the concepts above.
In triangle ABC, angle A = 30\u00B0, angle B = 45\u00B0, and side a = 10 cm. Using the Sine Rule, calculate the length of side c (AB) to one decimal place. (Type only the number, e.g., 19.3)
Review the concepts above.
A boat sails 30 km from port on a bearing of 040\u00B0, then turns and sails 40 km on a bearing of 130\u00B0. What is the overall bearing from the port to the boat's final position (to the nearest whole degree)? (Type only the number, e.g., 93)
Review the concepts above.
A ship sails from point X 8 km on a bearing of 045\u00B0 to point Y, then 6 km on a bearing of 135\u00B0 to point Z. Determine the bearing of line XZ from point X (to the nearest whole degree). (Type only the number, e.g., 82)
Review the concepts above.
In triangle PQR, side p = 12 cm, side q = 15 cm, and included angle R = 45\u00B0. Calculate side r to two decimal places using the Cosine Rule. (Type only the number, e.g., 10.70)
Review the concepts above.