Composition of approximations?!
Hello,
Here is the question I am stuck on:
For small x, sinx ~ x - (x^3)/6 and cosx ~ 1 - (x^2)/2 (~ means approx. = to)
find an approximation, ignoring powers of x > 3, for:
sin(x/2)cosx + sec2x
Their answer is:
1 + x/2+2x^2-13(x^3)/48
Now what I get stuck on is: on substituting the above sinx and cosx approxs. into the given expression, I always end up with a (1-2x^2) denominator, which I cannot eventually cancel, thus cannot produce the form of answer they give.
NB. This question is assumed to rely on only Core 1-->4 knowledge, not any Further Pure. Hence power series/Maclaurin and Taylor, shouldn't be applicable.
I suspect I am missing something pretty basic!!
Thanks in advance.
Here is the question I am stuck on:
For small x, sinx ~ x - (x^3)/6 and cosx ~ 1 - (x^2)/2 (~ means approx. = to)
find an approximation, ignoring powers of x > 3, for:
sin(x/2)cosx + sec2x
Their answer is:
1 + x/2+2x^2-13(x^3)/48
Now what I get stuck on is: on substituting the above sinx and cosx approxs. into the given expression, I always end up with a (1-2x^2) denominator, which I cannot eventually cancel, thus cannot produce the form of answer they give.
NB. This question is assumed to rely on only Core 1-->4 knowledge, not any Further Pure. Hence power series/Maclaurin and Taylor, shouldn't be applicable.
I suspect I am missing something pretty basic!!
Thanks in advance.
In order to eliminate 1/(1-2x^2) multiply top and bottom by (1+2x^2).
If you expand the denominator you then get (1 - 4x^4) = 1 as you can ignore terms in x^4,
Alternatively use the binomial theorem on (1 - 2x^2) ^ (-1) which leads to the same answer.
If you expand the denominator you then get (1 - 4x^4) = 1 as you can ignore terms in x^4,
Alternatively use the binomial theorem on (1 - 2x^2) ^ (-1) which leads to the same answer.
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