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Question
Find the maximum and minimum values of each of the following trigonometrical expression:
12 sin x − 5 cos x
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Solution
\[\text{ Let } f\left( x \right) = 12 \sin x - 5 \cos x\]
\[\text{ We know that }\]
\[ - \sqrt{{12}^2 + ( - 5 )^2} \leq 12 \sin x - 5 \cos x \leq \sqrt{{12}^2 + ( - 5 )^2}\]
\[ - \sqrt{144 + 25} \leq 12 \sin x - 5 \cos x \leq \sqrt{144 + 25}\]
\[ - 13 \leq 12 \sin x - 5 \cos x \leq 13\]
\[\text{ Hence the maximum and minumun values of }f\left( x \right) \text{ are 13 and - 13, respectively } .\]
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