A right triangle is one that has a 90° angle, and there can only be one in a triangle. If you have some configuration with no right angle, then you will have to generate a solution using a right angle which you introduce to the initial configuration, especially implicitly so, such as when dropping the normal is used in proving the Law of Cosines.
Sine and Cosine are very similar to one another, and they have characteristics of the right triangle and also the circle. They are ratios of a side to the hypotenuse, and functions of the acute angle. determining the opposite and adjacent sides.
SOH-CAH-TOA is the second most important thing in trigonometry (after Pythagoras, of course), which gives the relationships of the sides of the right angle triangle to the three (very common) functions, sine, cosine, and tangent in a three-syllable mnemonic.
Area of a circle: $$ \pi r^2 $$ where $r$ stands for the radius, or distance from the center of the circle to the edge. -- place circle pic w r, 2x2 grid -- And note that the area of the square which is $2r$ on the side, is $4r^2$ and that the circle covers roughly 3/4 of the square. Pi was first studied by Archimedes, who bounded the number $\pi$ by the sum of sides of a smaller polygon (inscribed) and a larger one (circumscribed). So the circumferance of the circle ($2\pi r$, or $\pi$ times the diameter) is between the inner polygon side length (less than) and the outer polygon side length (greater than). $$ Nl_{inner} \lt 2\pi r \lt Nl_{outer} $$ where $l$ is the distance between adjacent polygon vertices, and $N$ is the number of sides (Archimedes reportedly made this calculation for $N=96$).
To calculate the Taylor series, we first calculate the derivative of the sine function, which by plotting the two functions side by side, it can be seen that for any point the slope of the sine function is equal to value of the cosine function. And, that for any point of the cosine function, its slope is equal to negative the value of the sine function.
The derivative of sin(x) evaluated at $\pi/2$ is zero, and periodically, but its not zero everywhere, so it's not a zero function.