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

[decisive]

Homework Statement

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

Homework Equations

Given above (general Taylor poly eqn not allowed in proof)

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.

 

[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.

 

[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))

 

[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

 

[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.

 

[Dick]

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.

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

 

[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.