'Solve sin x + 4 cos x = 3'

Can anyone help me with the above problem? Trigonometric identities and formulae confuse me.

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I have tried doing this:

sin x = 3 - 4 cos x

divide all by cos x

then tan x = 3/cos x - 4

rearrange so 4 = 3/cos x - tan x

which is same as 4 = 3 sec x - tan x

square both sides

so 16 = 9 sec^2 x - tan^2 x

this is 16 = 9(tan^2x + 1) - tan^2 x

and 16 = 9tan^2x - tan^2x + 9

rearrange so 7 = 8 tan^2 x

7/8 = tan^2 x

so tan x= sqrt(7/8)

and x = arctan sqrt(7/8)

Is this correct?

If anyone could provide any help or solution to this problem, I'll be so grateful.

Thank you!

Herman