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F7 Trigonometric Functions
Students Srst encoutered cosine and sine in geometry where they studied right triangles and found that the ratio of side lengths only depend on the angles, not on the size of the triangle. This unit begins by recalling these ratios, focusing on the important fact that when the hypotenuse of the right triangle has unit length, the length of the legs can be expressed with cosine and sine. Students work extensively with the unit circle in the coordinate plane, studying right triangles with one vertex at the origin, one vertex on the circle, and one side along the -axis. Using the unit circle and these right triangles, they extend the deSnition of cosine and sine, Srst from acute angles to any angle with a measure between 0 and 360 and then, after making sense of negative angle measure using the unit circle, to any real number value.
Students view cosine and sine as functions of angle measure and graph these functions, relating the periodicity of the functions to the unit circle. Next, they recall another important concept from geometry, the idea of radian angle measure. Radian angle measure is ideally suited for measuring the distance an object in circular motion travels, such as a point on a spinning wheel. Students examine the cosine and sine functions with angle measure input in radians, observing their periodicity and gaining Tuency with measuring angles using radians. They apply radian angle measure to studying the motion of hands on a clock and graph the cosine and sine functions when the input is
measured radians.
Previously, students have examined diVerent transformations of graphs representing functions, including horizontal and vertical translations as well as reTections and vertical stretch. In this unit, students examine vertical translations and stretch, with vertical translation associated to the midline of a trigonometric function and the vertical stretch asociated with the amplitude. Students then investigate horizontal translations, recalling similar work with quadratic functions. Trigonometric functions have one additional parameter, namely the period, and students investigate the period of trigonometric functions both algebraically and geometrically.
Finally, students apply trigonometric functions to model diVerent situations. Some of these involve motion in a circle, such as examining a point on a Ferris Wheel or carousel. But there are also many periodic phenomena that are not directly associated with position on a circle. Some interesting examples which students study include tides, tourism data throughout the year, traUc also throughout the year, and the number of hours of daylight as it varies through the year in a particular location. These investigations all bring out diVerent aspects of the modeling cycle.
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Course Guide Algebra