Warm Up
The sine ratio describes the relationship between two sides of a right triangle. The sine of an acute angle in a right triangle is equal to the ratio of the side length opposite that angle to the hypotenuse length.
Two sine ratios for the same triangle are shown below.
Sometimes the sine ratio is written as 
or 
.
The sine ratio can be used to represent or determine unknown values in a right triangle.
Example 1
a. For the following triangle, determine sin p and sin q.
To determine sin p, start by labelling the diagram.
Substitute the known values into the sine ratio equation.
To determine sin q, start by labelling the diagram.
Substitute the known values into the sine ratio equation.
b. Determine the values of p and q.
Once you know the sine ratio, you can use the inverse sine feature on your calculator to determine an angle's measure. This is usually shown as sin-1 or arcsin on calculators.
Notice that the three angles in the triangle (p, q, and the right angle) add to 180°, just as expected.
A guide to using a TI83/84 calculator for solving equations that include a trigonometric ratio can be found in the Equipment Room or by clicking here.