Target Mathematics

1.17: Cotangent equation and secant identity

0606/12/O/N/17 — Question 11 · 14 marks

(a)
Solve 2cot⁡(ϕ+35∘)=52\cot(\phi + 35^{\circ}) = 5 for 0∘≤ϕ≤360∘0^{\circ} \le \phi \le 360^{\circ}.
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(b)(i)
Show that sec⁡θcot⁡θ+tan⁡θ=sin⁡θ\tfrac{\sec \theta}{\cot \theta + \tan \theta} = \sin \theta.
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(b)(ii)
Hence solve sec⁡3θcot⁡3θ+tan⁡3θ=−32\tfrac{\sec 3\theta}{\cot 3\theta + \tan 3\theta} = -\tfrac{\sqrt{3}}{2} for −π2≤θ≤π2-\tfrac{\pi}{2} \le \theta \le \tfrac{\pi}{2}, giving your answers in terms of π\pi.
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