First slide

Trigonometric equations

Question

$\cos 2 x + a \sin x = 2 a - 7 p o s s e s s e s a s o l u t i o n f o r :$

Moderate

A

$a l l a$
B

$a > 6$
C

$a < 2$
D

$a \in [2, 6]$

Solution

$\begin{array}{l} W e h a v e \cos 2 x + a \sin x = 2 a - 7 \\ \Rightarrow 1 - 2 \sin^{2} x + a \sin x = 2 a - 7 \\ \Rightarrow 2 \sin^{2} x - a \sin x + 2 a - 8 = 0 \\ \Rightarrow \sin x = \frac{a \pm \sqrt{a^{2} - 16 (a - 4)}}{4} \\ = \frac{a \pm (a - 8)}{4} \\ ⇒ \sin x = \frac{a - 4}{2} (o r) 2 \\ S i n c e \sin x \neq 2, w e h a v e \sin x = \frac{a - 4}{2} \\ Now - 1 \leq \sin x \leq 1 \\ ⇒ - 1 \leq \frac{a - 4}{2} \leq 1 \\ ⇒ 2 \leq a \leq 6 \end{array}$

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