Starting with defining $(\cos t, \sin t)$ from the unit circle, is it possible to derive the power series for $\sin(t)$:
$$\sin t = t – \frac{t^3}{3!} + \frac{t^5}{5!} – \dots$$
Note: I will be answering my question, I hope this doesn't offend anyone. If is any issue with the proof, I am grateful for any improvement.
Best Answer
Another method is using the inequality
$$\sin(x)\leq x$$
Integrate this between $0$ and $t$
$$-\cos(t)+\cos(0)\leq t^2/2$$ $$1-t^2/2\leq \cos(t)$$
Integrate this between $0$ and $x$:
$$x-x^3/3!\leq \sin(x)$$
Repeat this and you will get a lower and upper bound, which are just partial sums of the Taylor series. Note that you directly get the taylor series for $\cos(x)$.