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Range of Trigonometric Expressions

If α+β=π2, th...

Question

If $α + β = \frac{π}{2},$ then the minimum value of the expression $6 sin α - 8 sin β + 12$ is

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Solution

Given : $α + β = \frac{π}{2}$
$sin β = sin (\frac{π}{2} - α) = cos α$

Now,
$6 sin α - 8 sin β + 12 = 6 sin α - 8 cos α + 12$

We know that,
$6 sin α - 8 cos α \in [- \sqrt{6^{2} + 8^{2}}, \sqrt{6^{2} + 8^{2}}] \Rightarrow 6 sin α - 8 cos α \in [- 10, 10] \Rightarrow 6 sin α - 8 cos α + 12 \in [2, 22]$

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Range of Trigonometric Expressions

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