In a right-angled triangle

MATHEMATICS · Multi-part · GCSE · Mathematics · Geometry and Measures · Geometry and Measures

Problem

In a right-angled triangle ABCABC, right-angled at BB, the lengths of the sides are AB=5 cmAB = 5\text{ cm}, BC=12 cmBC = 12\text{ cm} and AC=13 cmAC = 13\text{ cm}. (i) Define sin⁡θ\sin \theta, cos⁡θ\cos \theta and tan⁡θ\tan \theta for an acute angle θ\theta in a right-angled triangle. (ii) Taking ∠ACB=θ\angle ACB = \theta, calculate the numerical values of sin⁡θ\sin \theta, cos⁡θ\cos \theta and tan⁡θ\tan \theta. (iii) Using the values obtained in part (ii), verify that sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1. Briefly explain, with reference to the triangle, why this relationship must always hold.

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