2 cos x + 3 sin 2x = 0

answer in degrees

If want just the approximated solutions in the interval from 0 to 360:

199.47

340.53

90

270

If you want all the approximated solutions:

199.47+360k

340.53+360k

90+360k

270+360k

Step-by-step explanation:

2 cos (x) + 3 sin (2x) = 0

First step: Use double angle identity for sin (2x). That is, use, sin (2x) = 2sin (x) cos (x).

2 cos (x) + 3*2sin (x) cos (x) = 0

2 cos (x) + 6sin (x) cos (x) = 0

Factor the 2cos (x) out, like so:

2cos (x) [ 1 + 3 sin (x) ]=0

In order for this product to be zero, we must find when both factors are 0.

2cos (x) = 0 or 1+3sin (x) = 0

Let's do 2cos (x) = 0 first.

2cos (x) = 0

Divide both sides by 2:

cos (x) = 0

So the x-coordinate is 0 on the unit at x=90 deg and x=270 deg (in the first rotation).

Let's do 1+3sin (x) = 0.

1+3sin (x) = 0

Subtract 1 on both sides:

3sin (x) = -1

Divide both sides by 3:

sin (x) = -1/3

Unfortunately this is not on the unit circle so I'm just going to take sin^-1 or arsin on both sides (this is the same thing sin^-1 or arsin).

x=arcsin (-1/3) = -19.47 degrees

So that means - (-19.47) + 180 is also a solution so 19.47+180=199.47.

And that 360+-19.47 is another so 360+-19.47=340.53.

So the solutions for [0,360] are

199.47

340.53

90

270

If you want all the solutions just add + 360*k to each line where k is an integer.