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Question
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
Solution
\[\text{ Let } f\left( x \right) = \sin x - \cos x + 1\]
\[\text{ We know that }\]
\[ - \sqrt{1^2 + ( - 1 )^2} \leq \sin x - \cos x \leq \sqrt{1^2 + ( - 1 )^2} for all x\]
\[ \Rightarrow - \sqrt{2} \leq \sin x - \cos x \leq \sqrt{2}\]
\[ \Rightarrow - \sqrt{2} + 1 \leq \sin x - \cos x + 1 \leq \sqrt{2} + 1\]
\[\text{ Hence maximum and minimum values of f(x) are } 1 + \sqrt{2} \text{ and } 1 - \sqrt{2} , \text{ respectively } .\]
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