Did you know that the ratio between the side of any triangle and the sine of the opposite angle is equal to the diameter of the triangle’s circumcircle? I didn’t! I just learned it today when researching the law of sines. All that time spent on the law of sines in high school, and no one ever bothered to tell me that in any triangle, not only are all the ratios between side lengths and sines of opposite angles equal to each other, they are also equal to something else interesting — namely, the diameter of the circumcircle!
Among other things, this means in particular that if you inscribe any angle in a circle with diameter one, the length of the chord it subtends is equal to the sine of the angle:
Nifty, eh? Want to see a proof? Here it is:
Rather than explain it in detail, I’ll just let you stare at it for a while, and leave a comment if you have questions. =) The one thing you need to remember from geometry that you might not remember is that an angle inscribed in a circle (like in the above picture) subtends an angle twice as large.