Writing A Trig Expression as an Algebraic Expression

[themadhatter1]

Homework Statement

Write the Trigonometric Expression as an algebraic expression.

cos(2arccos 2x)

Homework Equations

Probably the inverse properties, I'm not sure.

The Attempt at a Solution

I know I can rewrite this equation as.

u= arccos 2x

cos(2cos u=2x)

I can also say that the adjacent leg is 2x units long and the hypotenuse is 1 unit long. Then using the pythagorean theorm I can figure the opposite leg to be sqrt(1-4x²)

I'm not sure If this is necessary though can someone point me in the right direction? The 2 in front of the arccos is throwing me off because if that wasn't there I would just use the inverse property and cos(arccos 2x) would equal 2x.

 

[rock.freak667]

If u=cos^-1(2x) then you want to find cos(2u).

cos(2u)=cos²u-sin²u=2cos²u-1 = 1-2sin²u

and cos²u = (cosu)²

 

[themadhatter1]

I'm not sure I understand why you'd want to find cos(2u)

The answer is supposed to be 8x²-1 and that's the answer listed in the back of the book.

 

[rock.freak667]

I'm not sure I understand why you'd want to find cos(2u)

The answer is supposed to be 8x²-1 and that's the answer listed in the back of the book.

cos(2cos^-1(2x))

if you put u = cos^-1(2x), wouldn't cos(2cos^-1(2x)) become cos(2u)?

 

[themadhatter1]

Thanks, now I understand.

Once you have simplified it to 2cos²u-1 all you have to do is simplify it with the u in place

cos²(arcsin 2x)=2x

2(2x)²-1

8x²-1

Thanks!