Ch 2 · 6.3 Non-right-angled Triangles · O Level Mathematics 4024
Topic 2 of 3 in Trigonometry — O Level Mathematics 4024 — 58 lessons.
Introduction
Surveyors and navigators rarely get right angles. The sine rule and cosine rule extend trigonometry to every triangle, and tfrac12absin C gives its area without a perpendicular height. (Syllabus section 6.3; in IGCSE 0580 it is section E6.5.)
Key ideas — Labelling
Use capital letters for angles and the matching small letter for the side opposite: side a is opposite angle A.
🧑🏫 Teacher's explanation
Imagine you are looking at a map of three cities. To calculate the distance between them, you need a consistent way to name everything so you don't get lost. In trigonometry, we use a 'matching pair' system. If you name a corner (an angle) with a capital letter like A, the road leading directly across from it must be named with the small letter a.
📖 Definitions
🔍 Worked example — Easy
In triangle ABC, angle A = 40^°, side a = 10 cm, and side b = 12 cm. Identify which side is opposite angle A.
🔍 Worked example — Standard
A triangle has sides a=5, b=7, and c=8. Which side is opposite angle C?
🔍 Worked example — Exam-style
In triangle XYZ, side x=8 cm, side y=10 cm, and side z=12 cm. If you are using the Cosine Rule to find angle Y, which side must be used in the denominator of the formula cos Y = x^2 + z^2 - y^2/2xz?
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — The sine rule
a/sin A = b/sin B = c/sin C or sin A/a = sin B/b = sin C/c
🧑🏫 Teacher's explanation
Imagine you are a surveyor trying to measure the distance across a lake. You cannot walk across the water, but you can stand on the shore, measure the distance to a tree on the other side, and measure the angles between your line of sight and a baseline you have laid out on land. Because the triangle formed by your position, the tree, and the end of your baseline isn't a right-angled triangle, standard SOH CAH TOA won't work.
📖 Definitions
🔍 Worked example — Easy
In triangle ABC, side a = 8 cm, angle A = 40^°, and angle B = 70^°. Calculate the length of side b.
🔍 Worked example — Standard
In triangle XYZ, x = 12 cm, y = 15 cm, and angle X = 35^°. Find the two possible values for angle Y.
🔍 Worked example — Exam-style
In triangle PQR, p = 10 cm, q = 12 cm, and angle R = 50^°. Calculate the area of the triangle.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — The cosine rule
a^2 = b^2 + c^2 - 2bccos A cos A = b^2 + c^2 - a^2/2bc
🧑🏫 Teacher's explanation
Imagine you are building a triangular garden frame. If you know the lengths of two sides and the exact angle between them, the third side is 'locked' into place. The Cosine Rule is essentially a generalized version of Pythagoras' Theorem. While Pythagoras only works for right-angled triangles, the Cosine Rule adds a 'correction term', -2bccos A, that adjusts for the fact that the angle A might be anything other than 90^°.
📖 Definitions
🔍 Worked example — Easy
In triangle ABC, side b = 7 cm, side c = 10 cm, and angle A = 60^°. Calculate the length of side a.
🔍 Worked example — Standard
A triangle has side lengths a = 5 cm, b = 8 cm, and c = 10 cm. Find the largest angle in the triangle.
🔍 Worked example — Exam-style
In triangle XYZ, x = 12 cm, y = 5 cm, and z = 13 cm. Determine if the triangle is acute, right-angled, or obtuse.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Area
Area = tfrac12 absin C
🧑🏫 Teacher's explanation
In a right-angled triangle, finding the area is easy: you just multiply the base by the height and divide by 2. But what if the triangle is 'tilted' and doesn't have a 90^° angle? We can't easily see the vertical height.
📖 Definitions
🔍 Worked example — Easy
Calculate the area of a triangle where two sides are 8 cm and 5 cm, and the angle between them is 30^°.
🔍 Worked example — Standard
A triangle has sides of length 7 cm and 12 cm. The angle between these sides is 115^°. Find the area to 3 significant figures.
🔍 Worked example — Exam-style
In triangle XYZ, XY = 10 cm, YZ = 14 cm, and the area is 42 cm^2. Find the possible values of the angle XYZ.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — The ambiguous case
When finding an angle with the sine rule, sin B = k has two solutions between 0^° and 180^°: B and 180^° - B. Both are possible if the angles still add to less than 180^°; the question usually tells you whether the angle is obtuse.
🧑🏫 Teacher's explanation
Imagine you are standing at one corner of a triangle and you have a fixed length for one side and a fixed angle for another. If that side is 'swinging' like a pendulum, it might hit the base line at two different spots before it swings too far or too short. This is the 'ambiguous case'.
📖 Definitions
🔍 Worked example — Easy
In triangle ABC, a = 8 cm, b = 10 cm, and ∠ A = 40^°. Find the possible values of ∠ B.
🔍 Worked example — Standard
In triangle XYZ, x = 7 cm, y = 12 cm, and ∠ X = 35^°. Find the possible values of ∠ Y.
🔍 Worked example — Exam-style
In triangle PQR, p = 11 cm, q = 15 cm, and ∠ P = 50^°. Determine the two possible values for ∠ Q.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
a = 8 cm, b = 11 cm, C = 40^°. c^2 = 64 + 121 - 2(8)(11)cos 40^° = 50.18, so c = 7.08 cm. Area = tfrac12(8)(11)sin 40^° = 28.3 cm².
🔍 Worked example — Easy
In triangle ABC, side a = 8 cm, side b = 11 cm, and angle C = 40^°. Calculate the length of side c and the area of the triangle.
🔍 Worked example — Standard
In triangle XYZ, angle X = 50^°, side x = 10 cm, and angle Y = 70^°. Find the length of side y.
🔍 Worked example — Exam-style
In triangle ABC, side a = 7 cm, side b = 9 cm, and angle A = 40^°. Determine the possible values for angle B.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) r^2 = 81 + 36 - 108cos 110^° = 153.9, r = 12.4 cm. (2) b = 12sin 48^°/sin 65^° = 9.84 cm. (3) cosθ = 16 + 36 - 81/48 = -0.604, θ = 127.2^°. (4) tfrac12(10)(7)sin 150^° = 17.5 cm².