We propose numerical algorithms which can be integrated with modern computer algebra systems in a way that is easily implemented to approximate the sine and cosine functions with an arbitrary accuracy. Our approach is based on Taylor's expansion about a point having a form of $kp$, $k\in\mathbb{Z}$ and $p=\pi/2$, and being chosen such that it is closest to the argument. A full error analysis, which takes advantage of current computer algebra systems
in approximating~$\pi$ with a very high accuracy, of our proposed methods is provided. A numerical integration application is performed to demonstrate the use of algorithms. Numerical and graphical results are implemented by MAPLE.