Sine & Cosine Rule
Year 10 (IGCSE) 📐 Geometry & Measures Apply sine rule, cosine rule for non-right triangles.
📐 The Sine Rule
The sine rule applies to any triangle. Use it when you have a matched angle–side pair plus one more piece of information.
⚡ Sine Rule
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$Example: $a = 7$, $A = 30°$, $B = 50°$. Find $b$: $\frac{7}{\sin 30°} = \frac{b}{\sin 50°}$ → $b = \frac{7 \times \sin 50°}{\sin 30°} \approx 10.72$
📐 The Cosine Rule
Use the cosine rule when you know all three sides, or two sides and the included angle (the angle between them).
⚡ Cosine Rule
$$a^2 = b^2 + c^2 - 2bc\cos A$$$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$Example: $b=5$, $c=7$, $A=60°$. Find $a$: $a^2 = 25 + 49 - 2(5)(7)(0.5) = 39$ → $a \approx 6.24$
📐 Area Using the Sine Formula
Find the area of any triangle when you know two sides and the included angle.
⚡ Triangle Area
$$\text{Area} = \frac{1}{2}ab\sin C$$Example: $a = 8$ cm, $b = 11$ cm, $C = 35°$. Area $= \frac{1}{2}(8)(11)\sin 35° = 44 \times 0.574 \approx 25.2$ cm²
Use the sine rule for angle–side pairs. Use the cosine rule for SAS or SSS situations.
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