SOLUTION: Hi, can you please help me with this question Find the range of x in the interval [0, 2π) for which 8 sin x + cos x > 4. The answers are apparently 2arctan(0.2) and 2 arctan(3).

In your words

I think the word "range" should be "domain". Anyway:

, 



Then since ,  or 

, 

The terms on the left are trig expressions. To make the term 4 on 
the right become a trig expression also, we play the clever trick 
of multiplying the 4 by 1, then replacing the 1 by the familiar
Pythagorean trig identity :

We use the formulas for sine and cosine of 2y:

















One of those factors is positive and the other is negative. So either

 or 

Those would involve only tan(y) if we divided through by cos(y). We could 
do that as long as cos(y) is not 0.  Let's make sure cos(y) is not zero.

If cos(y) were 0, then  so let's substitute  in



<--false!

So cos(y) can't be 0. Therefore, we can divide by it. For the first case









Since tangent is a strictly increasing function,





Now since ,





That's the interval you were looking for.  If we considered the other case



it would lead to



which is a contradiction, so the domain is



Edwin