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## 'Building Approximations for Sin(x)' printed from http://nrich.maths.org/

We first encounter the function
sin(x) when using right-angled triangles, where sine of an angle is
defined to be the ratio of the length of the side opposite the
angle to the length of the hypotenuse. Clearly this ratio must give
a number which varies continuously between 0 and 1 as the angle
varies from 0 to 90 degrees .

However, your calculator will
give you a value of sin(x) for any value of x you care to choose,
no matter how big or small. Calculators do not do this by 'drawing
triangles'. Instead they approximate sin(x) by a polynomial
function. In this problem we shall investigate the properties of
these polynomials.
Look at an approximation by a cubic polynomial:

$$ \sin(x)\approx a + b x + c x^2+ d x^3\quad\quad\mbox{where }\,
a, b, c \,\mbox{ and } \, d \,\mbox{ are constants} $$ (where $x$
is in radians). By experimenting with the numerical values of
$\sin(x)$ what values would you suggest for the constants?

[Aside: Why is it not
possible to 'solve' for these constants?]

Extend your solution to 4th, 5th and higher order
polynomials.

Test the accuracy of your final solution over a range 0 to $\pi$
radians.

Why is it sufficient to work within this range?

Extension : Repeat this
challenge to find approximations for $\cos(x)$ and
$\log(1+x)$.

Discussion point : Why
does it make sense to be able to approximate $\sin(x)$ by a
polynomial? What other functions do you think would have polynomial
approximations? Do you think that all functions can be approximated
by polynomials?