In this unit students build an understanding of ratios in right triangles, which leads to naming cosine, sine, and tangent as trigonometric ratios.

Prior to beginning this unit, students will have considerable familiarity with right triangles and similarity. They learned to identify right triangles in grade 4. Students studied the Pythagorean Theorem in grade 8, and used similar right triangles to build the idea of slope. This unit builds on this extensive experience and grounds trigonometric ratios in familiar contexts.

Several concepts build throughout the unit. Students begin by using similar triangles to create a table of ratios of the side lengths in right triangles. At first, their table includes only the bottom rows of the table shown here. Taking the time to both build and use the table helps students construct a solid foundation before they learn the names of trigonometric ratios.

Students notice patterns between the columns for cosine and sine before they first hear the terms “cosine” and “sine.” In a subsequent lesson they investigate that relationship, proving the two ratios are equal for complementary angles. Finding the measures of acute angles in a right triangle follows a similar arc, where students first use the table to estimate and then in a subsequent lesson learn how to calculate an angle measure given the side measures by using arcsine, arccosine, and arctangent.

As students measure side lengths and compute ratios, there is an opportunity to discuss precision. In this unit, students will round side lengths to the nearest tenth and angle measures to the nearest degree in most cases. When students solve problems in context they grapple with whether or not their answer is reasonable, as well as the appropriate degree of precision to report.

Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.