Sine+Function

Lesson #9 - The Sine Function
Initial posting - July, 2011 Revision date - Fall 2017

** Purpose: ** This lesson introduces the sine function, which is used to find an unknown side length in a given right triangle. Note that basic knowledge of the sine function can and should be taught to middle school students. ** Essential Question: ** Given a right triangle, three known angle measurements and one known side length, how can the sine function and the Pythagorean theorem be used to calculate the missing side lengths? A Section of the Lesson Follows

A calculator can be used to find the sine of different angle measurements. Scientific calculators have more than one mode setting; so make sure that the mode is set to **degrees**. Some calculators require you to enter the sine button and then the angle’s degrees while others need the degree measurement entered first. Practice with your calculator on the following problems before completing Example 1. sin(20°) ≈ .342 sin(45°) ≈ .707 sin(60°) ≈ .866 sin(30°) = .500    It is important to round results to at least three decimal places when calculating trigonometric functions that need rounding, because the values are so close together. If the sines of 45°, 46°, 47°, and 48° are rounded to the tenths place they would all have the same value of 0.7. There are two additional trigonometric functions called cosine and tangent. They are commonly found in most high school geometry textbooks and are omitted in this series of lessons.

** Example 1: ** Triangle PQR has angle measurements 30°, 60°, 90°. The short side measures 20 units. Find the measurements of the other two sides The sine of angle P equals the ratio of RQ to PQ.
 * The sine function is a trigonometric function.

sin(angle) = opposite side length / hypotenuse length ||

Figure 9.1 ||
 * Note that sin(30°) = .5

Computation to find h: sin(30°) = 20/h .5 = 20/h .5 * h = 20 h = 40 units ||^  || Since we calculated the hypotenuse of triangle PQR as 40 units, we can now use the Pythagorean theorem to find the third side length. 1600 = 400 + PR^2 1200 = PR^2 SQRT(1200)= PR 34.6 units ≈ PR || fd 34.6 rt (180-30) fd 40 rt (180-60) fd 20 ||
 * ** Computation to find m(PR): ** ||  ** Logo Code **  ||  ** Completed graphic **  ||
 * 40^2 = 20^2 + PR^2

Figure 9.2 ||