sin x−sin 5x=sin 3x find the angle that satisfied the equestion


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Question Number 43346 by peter frank last updated on 10/Sep/18

$\sin x - \sin 5 x = \sin 3 x f i n d t h e a n g l e$ $t h a t s a t i s f i e d t h e e q u e s t i o n$
	Answered by tanmay.chaudhury50@gmail.com last updated on 10/Sep/18

	$s i n x = s i n 5 x + s i n 3 x$ $s i n x = 2 s i n 4 x . c o s x$ $s i n x = 2 (2 s i n 2 x c o s 2 x) c o s x$ $s i n x = 4 (2 s i n x c o s x) c o s 2 x c o s x$ $s i n x - 8 {s i n x c o s}^{2} x . c o s 2 x = 0$ $s i n x (1 - 8 {c o s}^{2} x . c o s 2 x) = 0$ $o n e s o l u t i o n$ $s i n x = 0 = s i n n Π$ $x = n Π$ $1 - 8 {c o s}^{2} x . c o s 2 x = 0$ $4 (2 {c o s}^{2} x) c o s 2 x = 1$ $4 (1 + c o s 2 x) c o s 2 x = 1$ $a = c o s 2 x$ $4 (1 + a) a = 1$ $4 a + 4 a^{2} - 1 = 0$ $4 a^{2} + 4 a - 1 = 0$ $a = \frac{- 4 \pm \sqrt{16 - 4 \times 4 \times (- 1)}}{2 \times 4}$ $a = \frac{- 4 \pm 4 \sqrt{2}}{2 \times 4}$ $a = \frac{4 \sqrt{2} - 4}{2 \times 4} a n d \frac{- 4 \sqrt{2} - 4}{2 \times 4}$ $a = \frac{\sqrt{2} - 1}{2} a n d \frac{- (\sqrt{2} + 1)}{2} \leftarrow t h i s n o t f e a s i b l e s o l n$ $1 ⩾ c o s 2 x ⩾ - 1$ $s o c o s 2 x c a n n o t b e \frac{- (\sqrt{2} + 1)}{2}$ $c o s 2 x = \frac{\sqrt{2} - 1}{2} \approx c o s 78^{o}$ $x \approx 39^{o}$ $r e c h e c k$ $s i n x - s i n 5 x = s i n 3 x$ $L H S$ $s i n 39^{o} - s i n (195^{o})$ $\approx 0.89$ $R H S$ $s i n (117^{o})$ $\approx 0.89$
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