Express \(8\sin\theta-7\cos\theta\) in the form \(R\sin(\theta-\alpha)\), where \(R>0\) and \(0^\circ<\alpha<90^\circ\).[3]
Hence find the maximum and minimum values of \(8\sin\theta-7\cos\theta\) and the corresponding values of \(\theta\) for \(0^\circ<\theta<360^\circ\).[3]
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Answer:(a) \(R=\sqrt{113}\), \(\alpha=\tan^{-1}\!\left(\dfrac78\right)\approx41.2^\circ\). (b) Maximum \(\sqrt{113}\) at \(\theta\approx131.2^\circ\); minimum \(-\sqrt{113}\) at \(\theta\approx311.2^\circ\).