In this lesson students find an angle given two side lengths of a right triangle. They learn to use a calculator to look up the angle corresponding to a ratio of sides and then apply that skill to a variety of problems and a context.

As students grapple with the idea of using sides to find angles, they will need to move from concrete to abstract. The arctangent, arccosine, or arcsine of a ratio tells them an angle measure in any right triangle with that particular ratio of side lengths. Students continue to reason abstractly and quantitatively (MP2) as they apply these ideas to the ladder scenario in an upcoming activity, especially if they do the measurements themselves.

Some calculators and texts use “arctan” and others use “.” The lesson summary uses the arc notation because it is clearer. Decide whether to introduce one or both notations in the Activity Synthesis of the Warm-up. In these materials , , and are used so students don’t confuse , which means the reciprocal of , with , which does not mean the reciprocal of tangent. If students' calculators (or an exam students will encounter) use and , then use that notation. Explain the potential for confusion, and clarify the difference between the notations.

This is the first lesson in which students encounter angle measurements produced by calculators using arccosine, arcsine, and arctangent. The calculator will provide an unreasonable number of decimal places in the output for most input ratios, and students have an opportunity to reason about how many decimal places to report. Because students typically measure angles using protractors that are precise to the nearest degree, these materials adopt the convention of rounding angles to the nearest whole number.