Help Understanding Trig Equation

[theintarnets]

Homework Statement

Find all possible solutions:
2cos²2θ = 1 - cos2θ

The Attempt at a Solution

I know all my arithmetic is correct, but when it comes to giving the answer, I'm not sure how to write it.
2cos²2θ + cos2θ - 1 = 0
(2cos2θ - 1)(cos2θ + 1) = 0
2cos2θ = 1
cos2θ = 1/2
2θ = ∏/3

cos2θ = -1
2θ = ∏

So θ is equal to ∏/6 and ∏/2
But the solutions are supposed to be:
∏/6 + ∏n
5∏/6 + ∏n
∏/2 + ∏n
And I don't understand why. Can someone explain it to me please?

 

[SammyS]

Homework Statement

Find all possible solutions:
2cos²2θ = 1 - cos2θ

The Attempt at a Solution

I know all my arithmetic is correct, but when it comes to giving the answer, I'm not sure how to write it.
2cos²2θ + cos2θ - 1 = 0
(2cos2θ - 1)(cos2θ + 1) = 0
2cos2θ = 1
cos2θ = 1/2
2θ = ∏/3

cos2θ = -1
2θ = ∏

So θ is equal to ∏/6 and ∏/2
But the solutions are supposed to be:
∏/6 + ∏n
5∏/6 + ∏n
∏/2 + ∏n
And I don't understand why. Can someone explain it to me please?

It's because the cosine function is periodic and is an even function.

 

[theintarnets]

Even as in cos(-x) = cos x? What do you mean by periodic? Thanks!

 

[SammyS]

cos(x) = cos(x + 2π) = cos(x + 4π) = cos(x + 2πk) , where k is an integer.

 

[theintarnets]

Ohhhhhh, I see, thank you! What is sin(x) then, because I know sin is odd, but I'm not sure how that looks.

 

[Bohrok]

An odd function means that f(-x) = -f(x). Graphically, f(-x) is just f(x) flipped across the x-axis or y-axis. Also, f(x) has rotational symmetry in that it's unchanged if you rotate it 180°.