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A simple rule for the simple sines
It is an often-used fact (in examples of elementary trigonometry and problems that involve it) that the sines of certain simple angles have simple expressions themselves. Specifically,
| sin(0) | = | 0 | ||
| sin(π/6) | = | sin(30°) | = | 1/2 |
| sin(π/4) | = | sin(45°) | = | √(2)/2 |
| sin(π/3) | = | sin(60°) | = | √(3)/2 |
| sin(π/2) | = | sin(90°) | = | 1 |
Furthermore, these statements about angles in the first quadrant can be reflected to handle similar common angles in the other three quadrants, and cosines. There is a common diagram illustrating this, considering sine and cosine as the coordinates of points on the unit circle.
When attempting to memorize these facts as one is expected to, I observed a pattern: each of the values may be expressed in the form √(x)/2.
| sin(0) | = | √(0)/2 | ||
| sin(π/6) | = | sin(30°) | = | √(1)/2 |
| sin(π/4) | = | sin(45°) | = | √(2)/2 |
| sin(π/3) | = | sin(60°) | = | √(3)/2 |
| sin(π/2) | = | sin(90°) | = | √(4)/2 |
I found this pattern to be quite useful. It does not, however, explain why those particular angles (quarters, sixths, eighths, and twelfths — but not tenths! — of circles) form this pattern of sines.
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