Question 1060955
You're doing great so far. The next step is to get everything in terms of either sine or cosine. I'm going to put everything in terms of cosine.


The Pythagorean Identity says that


sin^2(x) + cos^2(x) = 1


we can isolate sin(x) to get


sin(x) = sqrt(1 - cos^2(x))


where "sqrt" stands for "square root"


Now use substitution to go from this equation

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

to this one

cos( x ) + 2 = 3*sqrt(1 - cos^2(x))


the equation now has everything in terms of cosine

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Let's make
z = cos(x)
which means
z^2 = cos^2(x)


another bit of substituting turns
cos( x ) + 2 = 3*sqrt(1 - cos^2(x))
into
z + 2 = 3*sqrt(1 - z^2)
which is slightly easier to deal with

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Now solve for z


z + 2 = 3*sqrt(1 - z^2)
(z + 2)^2 = [3*sqrt(1 - z^2)]^2
z^2 + 4z + 4 = 9(1-z^2)
z^2 + 4z + 4 = 9-9z^2
z^2 + 4z + 4 - 9+9z^2 = 0
10z^2 + 4z - 5 = 0


After using the quadratic formula to solve for 'z', we get these approximate solutions


z = -0.93485
z = 0.53485


Note: I'm skipping the steps showing the quadratic formula to save space. Let me know if you need to see these steps or not. 


Recall that earlier we let
z = cos(x)


so if z = -0.93485, then
z = -0.93485
cos(x) = -0.93485
x = arccos(-0.93485)
x = 159.20392491748
x = 159.2
which is approximate of course


Similarly, if z = 0.53485, then
z = 0.53485
cos(x) = 0.53485
x = arccos(0.53485)
x = 57.6662606444783
x = 57.7
again which is approximate



So 
z = -0.93485 and z = 0.53485
lead to
x = 159.2 and x = 57.7
respectively in that order