Definition: In a right triangle, the sine of an acute angle is the length of the side opposite that angle divided by the length of the hypotenuse.
What does sine measure?
For an acute angle A, sin A = opposite / hypotenuse. The value depends only on the angle, not on the size of the triangle, because right triangles with the same acute angle are similar.
How do you use it?
To find a missing side, write an equation using sine, then solve. To find an angle, use the inverse sine key, written sin-1.
Example. A ramp rises 1.5 m and makes a 12 degree angle with the ground. The ramp length L satisfies sin 12 degrees = 1.5 / L. Since sin 12 degrees is about 0.2079, L = 1.5 / 0.2079, or about 7.2 m. To find an angle instead: if the opposite side is 9 and the hypotenuse is 15, sin A = 0.6, and A = sin-1(0.6), about 36.9 degrees.
What mistakes occur?
Using the adjacent side instead of the opposite one, and leaving the calculator in radian mode. Also, sine of an acute angle is always between 0 and 1, so a result above 1 signals an error. The label opposite depends on which angle you are using.