Mathematics

Mathematics

Problem

An acute angle θ\theta satisfies sin⁡θ−cos⁡θ=13\sin \theta - \cos \theta = \dfrac{1}{3}. a. (i) Prove that sin⁡θ+cos⁡θ=173\sin \theta + \cos \theta = \dfrac{\sqrt{17}}{3}. b. (ii) Hence determine the exact values of sin⁡θ\sin \theta and cos⁡θ\cos \theta. c. (iii) Obtain the value of tan⁡θ\tan \theta and verify that sin⁡2 ⁣θ+cos⁡2 ⁣θ=1\sin^{2}\!\theta + \cos^{2}\!\theta = 1 for the values found.

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