$cos 2^x + a sin x = 2 a - 7$ has a solution for what values of a?

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Solution

$c o s 2 x + a s i n x = 2 a - 7 1 - 2 s i n^{2} x + a s i n x = 2 a - 7 2 s i n^{2} x - a s i n x + (2 a - 8) = 0 f o r i t t o h a v e r e a l s o l u t i o n D \geq 0 (- a)^{2} - 4 \times 2 \times (2 a - 8) \geq 0 a^{2} - 16 a + 64 \geq 0 (a - 8)^{2} \geq 0 w h i c h i s v a l i d f o r a n y v a l u e o f a n o w - 1 ⩽ s i n x ⩽ 1 - 1 ⩽ (\frac{- (- a) \pm \sqrt{D}}{2 \times 2}) ⩽ 1 - 4 ⩽ a \pm | a - 8 | ⩽ 4 i f a > 8 t h e n - 4 ⩽ a + a - 8 ⩽ 4 - 4 ⩽ 2 a - 8 ⩽ 4 4 ⩽ 2 a ⩽ 12 2 ⩽ a ⩽ 6 (n o t p o s s i b l e) i f a < 8 t h e n - 4 ⩽ a - a + 8 ⩽ 4 - 4 ⩽ 8 ⩽ 4 (n o t p o s s i b l e) h e n c e f o r n o v l a u e o f a, e q u a t i o n w i l l h a v e s o l u t i o n$

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Similar questions

cos 2 x + a sin x = 2 a - 7

cos 2 x + a sin x = 2 a - 7

a

a sin x + cos 2 x = 2 a - 7

c o s 2 x + a s i n x = 2 a - 7

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