Question

`sin(2x)=2cos(2x)-1`, solve for `x` (`0<=x<=180`). Can somebody please solve this?

Answer 1

Solve sin 2x = 2cos 2x - 1

Ans: `18^@35 and 135^@09`

Explanation:

sin 2x - 2cos 2x = -1 (1)
Call tan a = sin a/(cos a) = 2 = tan 63.43 --> cos a = 0.45
Equation (1) -->
`sin 2x - (sin a/(cos a))cos 2x = -1`
sin 2x.cos a - sin a.cos 2x = -cos a (2)
Apply the trig identity: sin a.cos b - sin b.cos a = sin (a - b).
sin (2x - a) = -cos a = - 0.45
(2x - a) = - 26.74 and (2x - a) = 180 + 26.74 = 206.74.
a. 2x - 63.43 = - 26.74 --> 2x = -26.74 + 63.43 = 36.69 --> `x = 18^@35`
b. 2x - 63.43 = 206.74 --> 2x = 270.17 --> `x = 135^@09`
Check by calculator.
x = 18.35 --> 2x = 36.69 --> cos 2x = 0.60 --> sin 2x = 1.60.
0.60 = 1.60 - 1 . OK
x = 135.09 --> 2x = 270.17 --> cos 2x = 0.00 --> sin 2x = 1
0 = 1 - 1 = 0. OK