Hello! In this video, we will use a graphing utility to solve real-world problems involving trigonometric functions. Generally speaking, equations can be solved by simply graphing an equation in terms of the unknown variable, and graphing the desired result while looking for points of intersection.

For example, in Cincinnati, the number of hours of daylight on day x (where x is the number of days after January 1) is modeled by the function, f-of-x equals twelve plus two point eight three times sine of the quantity two-pi over three hundred sixty-five times the quantity of x minus eighty. We can determine which days of the year have about 10 hours of daylight by first graphing the function, and second by graphing our desired result. I am using desmos.com to complete my graphs, but a graphing calculator will work just as well (break). I have already adjusted the scale, and you can see the original function graphed here, and the desired result, graphed as the line y equals ten, here. We only need to look at the points of intersection between day zero and three hundred sixty five. The two times that the lines intersect occur at day thirty-four and day three hundred eight. This would be February 3rd and November 4th.

In our next example, we are reminded that the phases of the moon are measured by the ratio, or portion of the lunar disc that is lit. The angle, x, created by the sun, earth, and moon, with the earth at the vertex, as shown here, determines the ratio. This ratio can be calculated by the function g-of-x equals one half times the quantity of one minus cosine of x. We can use this function to determine the angles necessary to form a new moon, crescent moon, and full moon. Try using a graphing utility on your own to determine the correct angles now (break). You can see here that a zero degree angle forms a new moon, a sixty or three hundred degree angle forms a crescent moon and a one hundred eighty degree angle forms a full moon.

Finally, consider this last example. A ten foot wide highway sign is adjacent to a roadway. As a driver approaches the sign, the viewing angle changes. We can express the viewing angle, theta, as a function of the distance x between the driver and the sign (break). If we assume that a two degree viewing angle is necessary in order for the sign to be legible, at what distance does this first occur? Well, by graphing this function as well as our desired result (break), we can see this occurs at approximately two hundred eighty-six point four feet.

Keep in mind that this graphing technique for solving equations works for non-trigonometric functions too. However, graphing to find solutions can be difficult because graphs, especially by hand, can be difficult to read and interpret. The use of a graphing utility is essential for accuracy when solving equations. Consider using this method for other problems as you encounter them. Good luck!