Sine and Cosine

A right triangle is one that has a 90° angle, and there can only be one in a triangle.

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 angle. It's best that everyone be familiar with the Wikipedia page for the Sine function , because there's no sense in repeating good work.

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 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 $N=96$).

$ax$ is shorthand for $a$ times $x$, and by the distributive property of algebra: $cb=cax$ is $c$ times $a$ times $x$, where also I could point out that $b=ax$.

The Differential Series

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.

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