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Worked solution

Use the amplitude to find both extrema

Find the maximum and minimum values of 2cos⁡x−3sin⁡x2\cos x-3\sin x.

CJC · O Level revision · Example 22

Answer

Maximum 13\sqrt{13}; minimum −13-\sqrt{13}.

Compare your reasoning

Combine the terms in one cosine

2cos⁡x−3sin⁡x=Rcos⁡(x+α),R=22+32=13,α=tan⁡−1(3/2)≈56.3∘.\begin{aligned}2\cos x-3\sin x&=R\cos(x+\alpha),\\R&=\sqrt{2^2+3^2}=\sqrt{13},\\\alpha&=\tan^{-1}(3/2)\approx56.3^\circ.\end{aligned}

The source uses Rcos⁡(x+α)R\cos(x+\alpha) with R>0R>0 and acute α. Retain the exact phase; its printed 56.3° is rounded. Use consistent angle units for x and α.

Use the full cosine range

−1≤cos⁡(x+α)≤1 ⟹ −13≤2cos⁡x−3sin⁡x≤13-1\le\cos(x+\alpha)\le1\ \Longrightarrow\ -\sqrt{13}\le2\cos x-3\sin x\le\sqrt{13}

The source takes cosine=1 for the maximum and −1 for the minimum. With unrestricted real x, both values are attained.

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