Simple trigonometric inequality, yet no simple solution.

Hey guys!

I've recently encountered this problem and I would like to know if any of you could present a more elegant or simple solution to it.

My solution:

When

So for we have:

We sketch the graph, find solutions and then see what happens if

Re: Simple trigonometric inequality, yet no simple solution.

Hello, Doubled144314!

I will limit my answers to Quadrant 1.

You can generalize the results if you like.

We note that

We have: .

Multiply by

. . . .

. . . .

. . . . . . . . . .

There are two cases: .

Case (1): .

Case (2): .

. . . . . . . .

. . . . . . . .

Therefore: .