Homework Help: Finding Taylor(x) of sec(x) via sec(x)*cos(x)=1

1. May 18, 2010

decisive

1. The problem statement, all variables and given/known data
Calculate the taylor polynomial T7(x) of order 7 about x=0 for the function f(x)=sec(x)
Do not take derivatives of secx and sub into taylor formula. Let the required polynomial be T7(x)=1+a_2*x^2+a_4*x^4+a_6*x^6 As degree6 is the same as degree 7. Sub this into (cosx)(secx)=1 and solve for unknown coefficients. You can quote the taylor poly of cosx without proof. There should be no appearances of X^8 or higher powers in your work.

BTW those _ are subscripts XD so a subscript 2,4,6

2. Relevant equations
Given above (general Taylor poly eqn not allowed in proof)

3. The attempt at a solution
First of all I multiplied the taylor poly of cos x which is simply 1-x^2/2 +x^4/4! -x^6/6!

by the eqn given for T7 secx. This resulted in a string of x terms with powers multiples of 2 from 2 to 12. I then factorised in terms of the unknown coefficients. So i ended up with a polynomial =0 as the 1 term on the left and right cancelled. (secx*cosx=1 ) Then i made myself 3 eqns by subbing x=1,2,3 into the eqn. Resulting in 3 horrendous eqns with fraction coeffs of a_2 to a_6. I solved these simultaneously over the course of 2 hours to find that a_6=0 which is clearly wrong if you have seen the solution to secx elsewhere. (I used wolfram alpha) I also noticed that in the orignal eqn from multiplying secx and cos x that the coeffs of a_2 a_4 and a_4 were x^2*cosx x^4*cosx and x^6cosx respectively. Neither method has worked. I hope i have showed i have spent a depressingly long time on this problem, any sort of guide to the solution would be fantastic.

2. May 18, 2010

Dick

Just multiply (1-x^2/2 +x^4/4! -x^6/6!)*(1+a_2*x^2+a_4*x^4+a_6*x^6) and set it equal to 1, ignoring all terms with powers higher than 6. Then equate the total coefficient of x^2 to 0. What do you get for a2? Use that to solve for a4 by setting the coefficient of x^4 to zero. Finally find a6.

3. May 18, 2010

gabbagabbahey

Hmmm... why not show us what you got when you expanded

$$(1+a_2x^2+a_4x^4+a_6x^6)\left(1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}\right))$$

4. May 19, 2010

decisive

thanks a lot for the tip on finding the coefficient of x and not the coeficcient. I was misinterpreting the questions hint i suspect :D

5. May 20, 2010

Gib Z

Please refrain from helping this user in future. This is an ASSIGNMENT question from the MATH1901 course at the University of Sydney.

decisive has violated the Physicsforums guidelines.

6. May 20, 2010

Dick

I thought the whole purpose of the Homework Help section was to help students with assignment questions. Am I missing something?

7. May 20, 2010

Gib Z

Sorry I should have mentioned that this is not a normal homework assignment, but a hand in assignment that counts towards the final mark in the course. The student also signs a form stating that they did not receive help from other people, plagiarize, etc, although collaboration with other students doing the assignment is allowed.