Question 1

Paper 1, no calculatorHard6 marks

A circuit model represents the normalized voltage at phase ϕ\phi by v(ϕ)=sin⁡ϕv(\phi)=\sin\phi A controller operates at phase θ\theta where 0<θ<π20<\theta<\frac{\pi}{2} During calibration, the normalized voltage at phase 2θ2\theta is 12\frac12

(a)

Starting from the compound-angle identities, derive the double-angle identities for sin⁡2x\sin 2x cos⁡2x\cos 2x and tan⁡2x\tan 2x

[3]
(b)

Determine all possible exact values of tan⁡θ\tan\theta

[3]

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