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'Taking Trigonometry Series-ly' printed from https://nrich.maths.org/
Part1:
There are various approaches to this part of the problem, although
the answers in radians should look something close to $$
\sin(x)\approx x-\frac{1}{6}x^3\quad\quad \cos(x) =
1-\frac{1}{2}x^2 $$
One approach would be to start using well known properties of trig
functions. For example, we know that $a$ must be $0$ and $e$ must
be exactly 1 because $\sin(0)=0$ and $\cos(0) = 1$.
We can also use some shortcuts because we know that $\sin(x) =
-\sin(-x)$ and $\cos(x)=+\cos(-x)$. So, we can say that
$$
\sin(x)\approx bx +dx^3\quad\quad \cos(x) \approx 1+fx^2
$$
Then look at a very small value of $x$, say $0.0000001$. Unless the
other coefficients are very large, we can ignore then. This tells
us that, approximately
$$
b\approx \sin(0.0000001) \quad f\approx
\sqrt{\frac{\cos(0.0000001)-1}{0.0000001^2}}
$$
We could find a value of $d$ by trying some larger values of
$x$.
It should be clear that the numbers in radians are much smaller
(and nicer) than in degrees.
Part 2:
$ \frac{d^2 f}{dx^2}+x =0 $
Substitute $ f(x) = sinx, f ' (x) = cosx , f ''(x) = -sinx $
$ \frac{d^2 f}{dx^2} =-x $
$\frac{df}{dx}= cosx =\int -x = -\frac{x^2}{\frac{x^5}{5!}2} + c$
(sub cos0 = 1 to find c=1)
$f(x)=sinx = \int cosx = \int -\frac{x^2}{2} + 1 = -\frac{x^3}{3!}
+ x + c $ (sub sin(0) = 0 to find c=0)
$\int sinx = -cosx = \int-\frac{x^3}{6!} + x = -\frac{x^4}{4!} +
\frac{x^2}{2!} + c$ (-cos(0) = c =-1)
$\int -cos = -sinx = \int -\frac{x^4}{4!} + \frac{x^2}{2!} -1 =
-\frac{x^5}{5!} + \frac{x^3}{3!} -x$
$\int -sinx = cosx = \int -\frac{x^5}{5!} + \frac{x^3}{3!} -x =
-\frac{x^6}{6!} + \frac{x^4}{4!}-\frac{x^2}{2!} + 1 $
Hence cosx = $1 -\frac{x^2}{2!}+ \frac{x^4}{4!}
-\frac{x^6}{6!}$
And Sinx = $x - \frac{x^3}{3!} +\frac{x^5}{5!}$
Extension:
Infinite power series of sinx =$x - \frac{x^3}{3!} +\frac{x^5}{5!}
+ ... + \frac{x^n}{n!}$ for odd n
Infinite series of cosx =$1 -\frac{x^2}{2!}+ \frac{x^4}{4!}
-\frac{x^6}{6!} +... \frac{x^n}{n!} $for even n
Here we make a comparison of the evaluation of \frac{\pi}{2} by the
series approxomations and the actuall functions.
$sin(\frac{\pi}{2}) = 1$
approximation :$sin(\frac{\pi}{2}) = 1.004524856$
$cos (\frac{\pi}{2}) =0 $
appromimation: $cos (\frac{\pi}{2}) =-8.945230096 x 10^{-4}$
Calculators and computers do not store a list of results for each
and every possible function. A much more efficient method is to
store a polynomial expression for each funtion using the
taylor/maclaurin series.