Solving trig inequalities without using a graphing calculator

Question

So i have an inequality

cos x< sin2x

i can solve it to get that x >π/6 or x>5π/6

But the actual answer is π/6< x < π/2 or x> 5π/6.

The solution provided is by the use of a graphing calculator, and i need to get the <π/2 restriction by algebric manipulation rather than graphing. Is there any method which will allow me to accomplish this?

Answer 1

Consider the function $f(x) = \sin 2x-\cos x$. This function can be rewritten as:

$$f(x)=\cos x ( 2\sin x -1)$$

Find the roots of $f(x) = 0$ for $x\in [0,2\pi)$. This is simple to solve.

Now check where $f(x) > 0$.