Sine Function
(rev. 9/5/2000, 2/23/2002, 9/16/2002)
The sine function of an angle q, abbreviated sin q, is defined to be the positive or negative height (y-coordinate) of a point on the unit circle whose ray to the origin makes an angle of q with the positive x-axis. As a result of this definition, sin q attains the following values for various values of q:
|
Value of q |
Value of sin q |
| 0 | 0 |
| Quadrant I (i.e., q between 0 and p/2) | positive, but less than 1 |
| p/2 | 1 |
| Quadrant II (i.e., q between p/2 and p) | positive, but less than 1 |
| p | 0 |
| Quadrant III (i.e., q between p and 3p/2) | negative, between 0 and –1 |
| 3p/2 | –1 |
| Quadrant IV (i.e., q between 3p/2 and 2p) | negative, between 0 and –1 |
| 2p (or, in fact, any integral multiple of p) | 0 |
The ASTC ("All Students Take Calculus") rule can help you keep track of these relationships.
Another good way to keep the definition of the sine function clear in your mind is to use the SOHCAHTOA rule and observe that in a unit circle, H = 1.
If you want q to be in degrees, it is good to use a different letter of the alphabet and mark the degree symbol explicitly. For example, if t = 45, then sin2 tº = 1/2. If ÐJ = 90º, then sin J = 1. (In the second example, we would not say J = 90. However, it would be correct to say J = 90º or J = p/2.)
Also note: Never write the abbreviation sin all by itself. You must always write sin a, sin q, or whatever. In other words, you must always take the sine of something. To use the technical term, the sine function must always have an argument. This rule applies to the five other trig functions as well.