the general form is:
$$f(x) = \sum\limits_{n=0}^{N} \ a_n \ x^n $$
for a few specific functions:
square root:
$$ \begin{align} f(x) & \approx \sqrt{1+x} \quad \quad 0 \le x \le 1 \quad \quad N=4\\ a_0 & = 1.0 \\ a_1 & = 0.49959804148061 \\ a_2 & = -0.12047308243453 \\ a_3 & = 0.04585425015501 \\ a_4 & = -0.01076564682800 \\ \end{align} $$
base 2 logarithm:
$$ \begin{align} f(x) & \approx \log_2(1+x) \quad \quad 0 \le x \le 1 \quad \quad N=6\\ a_0 & = 0.0 \\ a_1 & = 1.44254494359510 \\ a_2 & = -0.7181452567504 \\ a_3 & = 0.45754919692582 \\ a_4 & = -0.27790534462866 \\ a_5 & = 0.121797910687826 \\ a_6 & = -0.02584144982967 \\ \end{align} $$
base 2 exponential:
$$ \begin{align} f(x) & \approx 2^x \quad \quad 0 \le x \le 1 \quad \quad N=4\\ a_0 & = 1.0 \\ a_1 & = 0.69303212081966 \\ a_2 & = 0.24137976293709 \\ a_3 & = 0.05203236900844 \\ a_4 & = 0.01355574723481 \\ \end{align} $$
sine (compute only odd-order terms):
$$ \begin{align} f(x) & \approx \sin\left(\frac{\pi}{2} x \right) \quad \quad -1 \le x \le 1 \quad \quad N=9 \\ a_0 & = 0.0 \\ a_1 & = 1.57079632679490 \\ a_2 & = 0.0 \\ a_3 & = -0.64596406188166 \\ a_4 & = 0.0 \\ a_5 & = 0.07969158490912 \\ a_6 & = 0.0 \\ a_7 & = -0.00467687997706 \\ a_8 & = 0.0 \\ a_9 & = 0.00015303015470 \\ \end{align} $$
cosine (use sine):
$$ \cos(\pi x) = 1 - 2 \sin^2 \left(\frac{\pi}{2} x \right) $$
arctangent (compute only even-order terms):
$$ \begin{align} \frac{x}{f(x)} & \approx \arctan(x) \quad \quad -1 \le x \le 1 \quad \quad N=8 \\ a_0 & = 1.0 \\ a_1 & = 0.0 \\ a_2 & = 0.33288950512027 \\ a_3 & = 0.0 \\ a_4 & = -0.08467922817644 \\ a_5 & = 0.0 \\ a_6 & = 0.03252232640125 \\ a_7 & = 0.0 \\ a_8 & = -0.00749305860992 \\ \end{align} $$
$$ \arctan(x) = \frac{\pi}{2} - \arctan\left( \frac{1}{x} \right) \quad \quad 1 \le x $$
$$ \arctan(x) = \frac{-\pi}{2} + \arctan\left( \frac{1}{-x} \right) \quad \quad x \le -1 $$