I'm trying to do an exercise that I really don't understand what or how to do it... or the part "Calculate sine and cosine according to formulas above with 7 expansion terms" do they mean that $n=7?$
Exe $\bf 1.5.2$ ($\bf 2$ points). Sine and cosine trigonometric functions can be presented as a series with the Taylor expansion \begin{align} \sin x &=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}\\ \cos x &=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}. \end{align} Calculate sine and cosine according to formulas above with $7$ expansion terms. The calculations should be performed in Command Window and should not exceed two lines.
The basic point is that you can calculate approximate values of infinite sums by calculating the sum of the first $M$ values for some $M$, i.e:
$$\sin(x) = \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1} \approx \sum_{n=0}^{M-1}\frac{(-1)^n}{(2n+1)!}x^{2n+1} $$
In your case you would take $M = 7$.
How good an approximation this is depends on the series and the value of $x$ - in general for this kind of power series the smaller $x$ is, the better an approximation it is. Increasing $M$ will also make a better approximation at the cost of increased computation time.