Radian measure- angle approximation expansion

Hello,

The first line is correct but I would stop there.

Note that if you were to avoid the fraction form, the second line should read (1-1/2 θ²)(1-3θ)^-1

Mmm, I just realised I’m meant to expand brackets if multiplying.

unfortunately I can’t stop here as that’s not what the question is looking for lol.

according to the textbook the answer should be
1+3theta+8.5theta² but there’s been many errors in this section in the textbook, which has left questioning a lot of the answers..

Edit: on that note of multiplying brackets, I found the answer to be 1x3theta-1/2theta² ignoring above

The textbook is correct. Youve truncated the expansion for sin(3x) at 3x which is correct as the next term is O(x^3), but when you do the series for
(1-3x)^(-1)
Youve not included the quadratic term and stopped at the linear term. This is incorrect as youre (part) multiplying it by 1 (well 1-x^2/2).

Edit - note you could have done this part
(1-x^2/2)/(1-3x)
using long division or matching coefficients. Though they probably expect a binomial expansion as you tried to do.

Let’s say theta=x

so, quadratic term for 1-3x?? I’m not sure that I follow.

if (1-3x)¹ where does this quadratic term come from?

it only exists in cos x≈ 1-1/2x²….

mmm, wait, I think you’ve just nudged a brain cell.

I didn’t expand 1-1/2x²

Edit: no there’s no expansion to be done here.
I’m so confused lol

As you can see by this example here. I’m simply using this concept the exact same way. I don’t know how my answer can be incorrect unless this example is wrong?

(1-3x)^(-1)
has an infinite number of terms in its series expansion. Newton would want to hang you along with the coin clippers. So
a + bx + cx^2 + dx^3 + ...
Youve only worked out a and b and truncated the rest. You need to include cx^2 as when you multiply it by (1-x^2/2), there are two contributions to the final quadratic so
cx^2 - ax^2/2

Doing it the other way suggested
1 - x^2/2 = (1-3x)(a + bx + cx^2)
then matching coefficients
constant: a = 1
linear: 0 = -3a + b, so b = 3
quadratic: -1/2 = c - 3b, so c = 8 1/2

Okay!!! After trying to give my brain a rest it clicked about having infinite number of terms!

I deserve the bullying from Newton lol.

I went back and re-addressed it and found the the solution.



I apologise for my stupid behaviour and I need a full days rest as I’ve clearly been over doing it!!