The purpose of this activity is to connect trigonometric ratios to the work students have done in previous lessons and in the Warm-up. Students learn the names of the trigonometric ratios and how to look them up on a calculator. To continue solidifying their conceptual understanding, students compare the calculator’s value to their work with the Right Triangle Table (MP6).

The purpose of this discussion is to compare two methods for finding an unknown side length in a right triangle: using a calculator and using the Right Triangle Table.

Invite students to share their new value for . (Solving the equation we wrote in the Launch gives .)

Ask students:

• “How does this answer compare to using the table?” (It’s the same.)
• “Which one is more trustworthy?” (All opinions are valid.)
• “Which one is more accurate?” (The calculator is much more accurate than the table for numbers between decade numbers. The calculator gives more decimal places, and it’s possible to solve without rounding to get a more precise answer.)

Inform students it is also okay to leave answers in the form , which would be exact.

Add these definitions to the class reference chart, and ask students to add them to their reference charts:

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. (Definition)

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. (Definition)

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. (Definition)

Students apply their new knowledge of trigonometric ratios to solve these problems. Since these problems ask for multiple side and angle measures, there is an increased opportunity for creativity in solving.

Monitor for students who solve multiple trigonometric equations versus those that apply the Pythagorean Theorem.

If students struggle to get started, prompt them to set up ratios to find the unknown sides. If students struggle to set up ratios, prompt them to identify what is known and what they are looking to find by annotating the diagram.

The purpose of this discussion is to contrast two methods for determining the final side length of a triangle: solving multiple trigonometric equations and applying the Pythagorean Theorem.

Ask students what the differences are between solving multiple trigonometric equations and applying the Pythagorean Theorem. Is one method easier? More accurate? (The Pythagorean Theorem is more familiar, so I prefer it. Both methods are accurate so long as you don't round too much.)

Call attention to the triangle with no marked right angle. Ask students, “How do you know you can use trigonometric functions with this triangle?” (By the Triangle Angle Sum Theorem, the third angle must be 90 degrees.)