Hey everyone, I'm doing some inequalities and I'm stuck with some trigonometric ones as I've never really encountered them before, some help with the following problem will be greatly appreciated.
Determine Θ, 0<=Θ<=2π so that (π = pi)
What I tried was to replace cos2Θ with and factorize till I end up with
From there on I'm not certain. I don't want to sinΘ >= -0.5 etc. because I'm afraid that will probably be wrong, How do I proceed? Thanks so much!
I assume this should read
.
Use the identity and this inequality becomes
Let and the inequality can be written as
, a Quadratic inequation. Quadratic inequalities are almost always easiest to solve by completing the square.
or
or .
or .
Case 1: Since for all real , that means the inequality is only true when .
.
Case 2: .
First, we can solve where
.
It should be clear that in the third quadrant of the unit circle, as your angle increases, the vertical distance gets greater in magnitude (so more negative). So any will make . Also, in the fourth quadrant, as your angle increases, the vertical distance gets smaller in magnitude (so less negative). So any will have . These are the values we disregard.
So finally, we have the solution .