Trig Identity: Solving Cos(x/2) = 1/2 and the Proper Use of Plus or Minus

[Miike012]

Cos(x/2) = 1/2

Trig Identity: Cos(x/2)= +- (1+Cosx/2)^(1/2) ..How do you know wheather to use the plus or minus of (1+Cosx/2)^(1/2) ? Do you only use the positive one because of the positive 1/2?
Anyways...

((1+Cosx)/2)^(1/2) = 1/2

(1+Cosx)/2 = 1/4

1+Cosx = 1/2

Cosx = -1/2

Ref Angle is pi/3 --> Cos is neg in Quad 2 and 3
Thus x Must be 2pi/3 and 4pi/3

However... in the back of the book it says the answers are 2pi/3 and 10pi/3...
How did I not come up with 10pi/3? what did I do wrong?

 

[Mentallic]

This isn't the way you want to solve the problem. Let y=x/2, solve for y, then substitute back to solve for x.

 

[Apphysicist]

Why don't you just Arccos both sides from the beginning? (I assume you capitalized Cos to mean the principal cosine)

 

[gb7nash]

Why don't you just Arccos both sides from the beginning? (I assume you capitalized Cos to mean the principal cosine)

This will only give you one value of x. However, we want all possibilities for x.

 

[Zach Knight]

Apphysicist and Mentallic are both right on this one. Taking the arccos first will give you two answers since cos is positive in two quadrants. Not quite sure what gb7nash is talking about.

 

[gb7nash]

Edit:

I just reread the original post. You're right.

If no info was given besides the equation though, arccos would only give you one value. To find all other values though:

{arccos + n*pi | n <- Z}