Mathematics • Year 10 • Unit 3 • Lesson 16
Cosine Rule in the Real World
Apply the cosine rule to real Australian contexts — a cricket pitch diagonal, a Sydney sailing course, a triangular paddock, a roof rafter, and a bush-walking route. Decide SAS vs SSS, write the right form, and check whether the answer is reasonable.
1. Word problems
For each problem: (i) sketch a triangle with all known information labelled, (ii) decide whether it is SAS or SSS, (iii) write the correct cosine-rule form, (iv) calculate. Round lengths to 1 dp and angles to the nearest 0.1°.
1.1 — Sailing leg at Pittwater. A yacht sails from a marker for 4.5 km on one course, then turns through an external angle of 65° and sails 3.2 km. The internal angle of the triangle (between the two legs) is therefore 180° − 65° = 115°. How far is the yacht from the original marker in a straight line? 3 marks
1.2 — Triangular paddock at Dubbo. A farmer surveys a paddock whose three sides measure 240 m, 310 m, and 420 m. He needs the angle at the corner where the two shorter sides (240 m and 310 m) meet, so he can install a gate. Find that angle. 3 marks
1.3 — Roof rafter on a Brisbane Queenslander. A symmetric A-frame roof has two rafters of length 5.4 m meeting at an apex angle of 70°. Find the width of the roof at the wall plate (i.e. the base of the triangle). 3 marks
1.4 — Bush-walking shortcut at Katoomba. Three lookouts P, Q, R form a triangle on a topographic map. PQ = 1.2 km, QR = 0.9 km, PR = 1.6 km. A ranger needs the angle at Q (the corner of the path she will turn through if she walks P → Q → R). Find ∠PQR. 3 marks
1.5 — Cricket ground diagonal. A boundary marker at the SCG is 65 m from the centre wicket and 78 m from a corner camera tower. The angle between these two sight-lines (at the marker) is 105°. How far apart are the centre wicket and the camera tower? 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A classmate says: "If you have two sides and any angle, just use the cosine rule." Explain in 4-6 sentences (i) why this is wrong when the angle is not between the two known sides, (ii) which rule should be used instead in that case, and (iii) what specific pattern of information (SAS / SSS / SSA / AAS) actually triggers the cosine rule from Lesson 16. Reference the words "included angle" somewhere in your answer.
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What I'll revisit before next class:
1.1 — Sailing leg (a = 4.5, b = 3.2, C = 115°)
SAS. c² = 4.5² + 3.2² − 2(4.5)(3.2) cos 115° = 20.25 + 10.24 − 28.8 × (−0.4226) = 30.49 + 12.17 = 42.66.
c = √42.66 ≈ 6.5 km. The obtuse angle makes c larger than √(4.5² + 3.2²) ≈ 5.5 km.
1.2 — Dubbo paddock (sides 240, 310, 420)
SSS. cos C = (240² + 310² − 420²) / (2 × 240 × 310) = (57 600 + 96 100 − 176 400) / 148 800 = −22 700 / 148 800 ≈ −0.1526.
C = cos⁻¹(−0.1526) ≈ 98.8°. Obtuse (as expected — 420 m is the longest side, so it faces the largest angle, and 240² + 310² < 420², so that angle exceeds 90°).
1.3 — Roof rafter (a = b = 5.4, apex C = 70°)
SAS. c² = 5.4² + 5.4² − 2(5.4)(5.4) cos 70° = 29.16 + 29.16 − 58.32 × 0.3420 = 58.32 − 19.95 = 38.37.
c = √38.37 ≈ 6.2 m. (The roof is about 6.2 m wide at the wall plate.)
1.4 — Katoomba lookouts (∠PQR)
SSS. At Q, the two sides are QP = 1.2 km and QR = 0.9 km, opposite side PR = 1.6 km.
cos Q = (1.2² + 0.9² − 1.6²) / (2 × 1.2 × 0.9) = (1.44 + 0.81 − 2.56) / 2.16 = −0.31 / 2.16 ≈ −0.1435.
∠PQR = cos⁻¹(−0.1435) ≈ 98.3°.
1.5 — SCG sight-lines (a = 65, b = 78, C = 105°)
SAS. c² = 65² + 78² − 2(65)(78) cos 105° = 4 225 + 6 084 − 10 140 × (−0.2588) = 10 309 + 2 624 = 12 933.
c = √12 933 ≈ 113.7 m. (Obtuse-angle case, so the opposite side is longer than √(65² + 78²) ≈ 101.5 m.)
2.1 — Explain your thinking (sample response)
The claim is not always correct. The cosine rule from Lesson 16 needs the angle to be the included angle — the one sitting between the two known sides. If the angle is opposite one of the known sides instead (the SSA case), the cosine rule cannot be applied directly because the formula c² = a² + b² − 2ab cos C expects C to be between a and b. In that situation the correct tool is the sine rule, which may give an ambiguous case (two possible triangles). So the patterns that actually trigger the cosine rule from Lesson 16 are: SAS (two sides and the included angle → find the third side) and SSS (all three sides → find any angle).
Marking: 1 for spotting the SSA exception, 1 for naming "included angle", 1 for naming the sine rule as the correct alternative, 1 for stating both SAS and SSS as the valid triggers.