Algebraic Expression for cos(arctan(x)/3)

Date: 8/15/96 at 15:51:32
From: Anonymous
Subject: Algebraic Expression for cos(arctan(x)/3)

We know that cos(arctan(x)) equals 1/sqr(x^2+1). Likewise, 
cos(arctan(x)/2) can be written in an algebraic manner. 
The problem that I have been trying to solve for quite a 
while is to find an algebraic expression for cos(arctan(x)/3), 
in order to get rid of the trigonometric operands. 

Thanks in advance for trying to help me out.

Arno de Puij, the Netherlands

Date: 10/29/96 at 15:3:59
From: Doctor Ceeks
Subject: Re: Algebraic Expression for cos(arctan(x)/3)

Hi,

As far as I know, you will have to use the formula for the solutions 
to a cubic, which are algebraic, to get what you want.

Using cos(x)= 4cos(x/3)^3 - 3cos(x/3), you can reduce cos(arctan(x)/3) 
by noting that cos(arctan(x))=1/root(x^2+1). Then you solve the cubic:

1/root(x^2+1) = 4y^3 - 3y

for y.

You can find the formula for the roots of a cubic equation at the
site:

 http://mathforum.org/dr.math/problems/3degree.html   

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/