Expert: Paul Klarreich - 10/14/2007

Question
Hi, I'm trying to simplify (1+cosx)^(1/2). I hope that makes sense I don't know how to type the mathematical notation.

Using cosx = cos2(x/2) I have got as far as (2(cos(x/2)^2))^1/2, which gives the same result on graphic calculator, but trying to square root the 2 terms gives (root2)cos(x/2) or (2^(1/2))(cos(x/2)). This looks much cleaner on paper but I don't know how to type it out.

Anyway using a graphics calculator gives a different result, it has the same period, phase, and amplitude but my simplified result has negative values, and is odd symetrically, whereas the original function gives only positive values, and is even.

Why does square rooting the 2 terms not work?

Answer
Questioner:   eddy
Category:  Advanced Math
Private:  No
 
Subject:  Mathematical Models
Question:  Hi, I'm trying to simplify (1+cosx)^(1/2). I hope that makes sense I don't know how to type the mathematical notation.

Using cosx = cos2(x/2) I have got as far as (2(cos(x/2)^2))^1/2, which gives the same result on graphic calculator, but trying to square root the 2 terms gives (root2)cos(x/2) or (2^(1/2))(cos(x/2)). This looks much cleaner on paper but I don't know how to type it out.

Anyway using a graphics calculator gives a different result, it has the same period, phase, and amplitude but my simplified result has negative values, and is odd symetrically, whereas the original function gives only positive values, and is even.

Why does square rooting the 2 terms not work?

..................................
Hi, Eddy,

You are on the right track, I think, and there is no real difficulty here.  You are using the 'half-angle' trick.  [When you reach Calc II, that's what you will call it.]
            1 + cos x
cos^2(x/2) = ----------
                2
or:

1 + cos x = 2 cos^2(x/2)

Now you want the square root, and the assumption is that it means PLUS sqrt(...).  In that case, you are supposed to write:

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

(2 cos^2(x/2))^1/2 =

(2)^1/2 ( cos^2(x/2) )^1/2 =

sqrt(2) ABS(cos(x/2)).   << THAT'S WHAT YOU MUST PUT INTO YOUR CALCULATOR

or, using other notation for absolute value:

sqrt(2)  | cos(x/2) |

The difficulty you are having is that:

1. Your basic algebra teacher told you that   sqrt(x^2) = x.
2. She LIED.  sqrt(x^2) = | x |, not x.
3. But don't get angry at her.  You were too young to understand, so she over-simplified.