Find-the-values-of-x-in-the-range-0-to-360-for-which-sin-3x-sin-x-2cos-2x-1-

Question Number 13903 by tawa tawa last updated on 24/May/17

Find the values of x in the range 0 ° to 360 ° for which

\sin (3 x) \sin (x) = 2 \cos (2 x) + 1

Commented by myintkhaing last updated on 26/May/17

P l e a s e i n t h e r a n g e 0 ° t o 360 ° m e a n s

0 ° < x < 360 ° o r 0 ° ⩽ x ⩽ 360 ° ? ?

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/May/17

(3 s i n x - 4 {s i n}^{3} x) s i n x = 2 (1 - 2 {s i n}^{2} x) + 1

s i n x = t \Rightarrow 3 t^{2} - 4 t^{4} = - 4 t^{2} + 3 \Rightarrow

4 t^{4} - 7 t^{2} + 3 = 0 \Rightarrow t^{2} = 1, \frac{3}{4} \Rightarrow t = s i n x = \pm 1, \pm \frac{\sqrt{3}}{2}

s i n x = 1 \Rightarrow x = 2 k π + \frac{π}{2}

s i n x = - 1 \Rightarrow x = 2 k π + \frac{3 π}{2}

s i n x = \frac{\sqrt{3}}{2} \Rightarrow x = 2 k π + \frac{π}{3}, 2 k π + \frac{2 π}{3}

s i n x = - \frac{\sqrt{3}}{2} \Rightarrow x = 2 k π + \frac{4 π}{3}, 2 k π + \frac{5 π}{3}

\Rightarrow x = \frac{π}{3}, \frac{π}{2}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{3 π}{2}, \frac{5 π}{3} \in (0, 2 π) .

Commented by tawa tawa last updated on 25/May/17

God bless you sir .

Commented by ajfour last updated on 25/May/17

i t h i n k x = \frac{2 π}{3}, \frac{5 π}{3} a l s o q u a l i f y

i n t h e g i v e n r a n g e .

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/May/17

y o u a r e r i g h t s i r . t h a n k s .

Answered by ajfour last updated on 25/May/17

2 \sin (3 x) \sin (x) = 4 \cos (2 x) + 2

\cos (2 x) - \cos (4 x) = 4 \cos (2 x) + 2

\cos (4 x) + 3 \cos (2 x) + 2 = 0

2 \cos^{2} (2 x) - 1 + 3 \cos (2 x) + 2 = 0

l e t t = \cos (2 x), t h e n

2 t^{2} + 3 t + 1 = 0

(2 t + 1) (t + 1) = 0

t = - \frac{1}{2}; t = - 1

\Rightarrow \cos (2 x) = - \frac{1}{2} w h e r e 0 ⩽ 2 x ⩽ 4 π

2 x = π \pm \frac{π}{3}, 3 π \pm \frac{π}{3}

x = \frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{5 π}{3}

a n d i f \cos (2 x) = - 1

2 x = π, 3 π

x = \frac{π}{2}, \frac{3 π}{2}

h e n c e i f 0 ⩽ x ⩽ 2 π, t h e n

x = \frac{π}{3}, \frac{π}{2}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{3 π}{2}, \frac{5 π}{3} .

Commented by mrW1 last updated on 25/May/17

y e s, t h e r e a r e 6 s o l u t i o n s i n t h e

r a n g e (0, 2 π)

Commented by ajfour last updated on 25/May/17

t h a n k s .

Facebook Tweet Pin