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Trigonometric Ratios of Multiple and Submultiple Angles — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Ratios of Multiple and Submultiple Angles1 Mark
Question
Write the maximum and minimum values of $3\cos\text{x}+4\sin\text{x}+5.$
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Answer

Let $\text{f(x)}=3\cos\text{x}+4\sin\text{x}+5$ We know that, $-\sqrt{3^2+4^2}\le3\cos\text{x}+4\sin\text{x}\le\sqrt{3^2+4^2}$ $\Rightarrow-\sqrt{9+16}\le3\cos\text{x}+4\sin\text{x}\le\sqrt{9+16}$ $\Rightarrow-\sqrt{25}\le3\cos\text{x}+4\sin\text{x}\le\sqrt{25}$ $\Rightarrow-5\le3\cos\text{x}+4\sin\text{x}\le5$ $\Rightarrow-5+5\le3\cos\text{x}+4\sin\text{x}+5\le5+5$ $\Rightarrow0\le3\cos\text{x}+4\sin\text{x}+5\le10$ Hence, minimum and maximum value of $3\cos\text{x}+4\sin\text{x}+5$ are 0 and 10 respectively.

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