Question

Solve on the interval 0 ≤ θ < 2π
cos(2θ-π/2)=-1

Answer 1

Don't I just love messing with trigonometric identities :D
First of all... what is
cos(-θ) equal to?

Answer 2

Well, I'll let this one go...
cos(-θ) = cos θ
Agree?

Answer 3

Yeah..

Answer 4

So...
\[\Large \cos\left(2\theta-\frac{\pi}{2}\right)=\cos\left[-\left(2\theta-\frac{\pi}{2}\right)\right]=\cos\left(\frac{\pi}{2}-2\theta\right)\]

Answer 5

Wait why does it become negative?

Answer 6

It didn't become negative.. I put in a negative sign.
You did agree that
cos(-x) = cos(x)
So what I did was turn the 2θ-π/2 into π/2-2θ And because
cos(-x) = cos(x)
that doesn't change the value.

Answer 7

The reason I did this is because we have another useful identity...
\[\huge \cos\left(\frac{\pi}{2}-\alpha\right)=\sin(\alpha)\]

Answer 8

Oh... haha okay

Answer 9

So...\[\huge \huge \cos\left(2\theta-\frac{\pi}{2}\right)=\cos\left(\frac{\pi}{2}-2\theta\right)=\sin(2\theta)\]
And I think you'll agree that that's much simpler.

Answer 10

So it becomes sin(2θ)=-1?

Answer 11

that it does :)