Prove that the equation $c o s 2 x + a s i n x = 21 - 7$ possesses a solution if $2 \leq a \leq 6$

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Solution

$c o s 2 x + a s i n x = 21 - 7$
$\Rightarrow 2 s i n^{2} x + a s i n x + (21 - 8) = 0 [∵ c o s 2 x = 2 s i n^{2} x - 1]$
$\Rightarrow s i n x = \frac{a \pm (a - 8)}{4}, = \frac{a - 4}{2}, 2 - 1 \leq s i n x \leq 1$
$∴$ Given equation has solution of $2 \leq a \leq 6$ .

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