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Question
Find the maximum and minimum values of each of the following trigonometrical expression:
12 cos x + 5 sin x + 4
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Solution
\[\text{ Let } f(x) = 12 \cos x + 5 \sin x + 4\]
\[\text{ We know that }\]
\[ - \sqrt{{12}^2 + 5^2} \leq 12 \cos x + 5 \sin x \leq \sqrt{{12}^2 + 5^2} for all x\]
\[ \Rightarrow - \sqrt{169} \leq 12 \cos x + 5 \sin x \leq \sqrt{169}\]
\[ \Rightarrow - 13 \leq 12 \cos x + 5 \sin x \leq 13\]
\[ \Rightarrow - 9 \leq 12 \cos x + 5 \sin x + 4 \leq 17\]
\[\text{ Hence, the maximum and minimum vaues of }f\left( x \right) \text{ are 17 and - 9, respectively } .\]
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