Lesson 6

Working with Trigonometric Ratios

Lesson Narrative

In this lesson students learn the names of the trigonometric ratios they have been using in the right triangle table and how to look them up on the calculator. Cosine is the ratio of the length of the adjacent leg to the length of the hypotenuse for a given acute angle in a right triangle. Sine is the ratio of the length of the opposite leg to length of the hypotenuse. Tangent is the ratio of the length of the opposite leg to length of the adjacent leg.

The decision to write the trigonometric ratios in the order “cosine, sine, tangent” is purposeful. Because cosine represents the \(x\)-coordinate in the unit circle, while sine represents the \(y\)-coordinate, tables with cosine first correctly correspond to the \((x,y)\) coordinates. This will not align with the SOHCAHTOA mnemonic, but it is not expected that students memorize these definitions. They can use their reference chart or the right triangle table for reference at any time.

Trigonometric functions (which will be studied more generally in a future course) have inputs of angle measures and outputs of ratios. Right now, students only know about angles measured in degrees, so they are only expected to interpret the cosine, sine, or tangent of an angle with the assumption that the angle is measured in degrees. People use the notation to mean a variety of things. Say angle \(A\) has a measure of 23 degrees. We might write \(\sin(23)\) or \(\sin(A)\). We interpret \(\sin(A)\) to mean sine of the measure of angle \(A\). Students don’t need this level of nuance, but be prepared to explain the different interpretations of trigonometric functions if it comes up.

A calculator gives far more decimal places for the value of a trigonometric ratio than it makes sense to use. Since the outputs of sine and cosine only range from zero to one, for inputs from zero to 90 degrees, rounding to the tenths place would also not make sense. It is important that students attend to precision (MP6) as they move from a table rounded to three decimal places to choosing how to use the very long decimal provided by a calculator. In this lesson students will discuss rounding and its effects in general. In future lessons students will discuss appropriate levels of precision based on given information.

Learning Goals

Teacher Facing

  • Calculate side lengths in right triangles using cosine, sine, and tangent (using words and other representations).

Student Facing

  • Let’s solve problems using cosine, sine, and tangent.

Required Materials

Required Preparation

Students need scientific calculators to evaluate trigonometric ratios. Make sure to set them to degree mode or prepare to instruct students how to change the mode.

Prepare additional copies of the Blank Reference Chart blackline master (double sided, 1 per student). Students can staple the new chart to their full ones as they will need to continue to refer to the whole packet.

Learning Targets

Student Facing

  • I can use cosine, sine, and tangent to find side lengths of right triangles.

CCSS Standards

Building On

Addressing

Glossary Entries

  • cosine

    The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse. In the diagram, \(\cos(x)=\frac{b}{c}\).

  • sine

    The sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse. In the diagram, \(\sin(x) = \frac{a}{c}.\)

  • tangent

    The tangent of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg. In the diagram, \(\tan(x) = \frac{a}{b}.\)

  • trigonometric ratio

    Sine, cosine, and tangent are called trigonometric ratios.