Click here to purchase the entire book in PDF format. Radians Once upon a time, someone discovered that there is a relationship between the radius of a circle and its circumference. It turned out that, no matter how big or small the circle, the circumference was equal to the radius multiplied by 2 and multiplied again by the number 3.141592645... That number was given the name ``pi'' (written $\pi$) and people have been fascinated by it ever since. In fact, the number does't stop where I said it did - it keeps going for at least a billion places without repeating itself... but 9 places after the decimal is plenty for our purposes. So, now we have a new little equation: $$\textrm{circumference} = 2 * \pi * r$$ (2.12) where r is the radius of the circle and $\pi$ is 3.141592645... Normally we measure angles in degrees where there are $360^{\circ}$ in a full circle, however, in order to measure this way, we need a protractor to tell us where the degrees are. There's another way to measure angles using only a ruler and a piece of string... Let's go back to the circle above with a radius of 1. Since we have the new equation, we know that the circumference is equal to $2\pi r$ - but $r=1$, so the circumference is $2\pi$ (say ``two pi''). Now, we can measure angles using the circumference - instead of saying that there are $360^{\circ}$ in a circle, we can say that there are $2\pi$ radians. We call them radians becase they're based on the radius. Since the circumference of the circle is $2\pi r$ and there are $2\pi$ radians in the circle, then 1 radian is the angle where the corresponding arc on the circle is equal to the length of the radius of the same circle. Using radians is just like using degrees - you just have to put your calculator into a different mode. Look for a way of getting it into ``RAD'' instead of ``DEG'' (RADians instead of DEGrees). Now, remember that there are $2\pi$ radians in a circle which is the same as saying 360 degres. Therefore, $180^{\circ}$ which is half of the circle is equal to $\pi$ radians. $90^{\circ}$ is $\frac{\pi}{2}$ radians and so on. You should be able to use these interchangeably in day to day conversation. This will impress your friends and strangers immensely.