## Trigonometry for Solving Problems

- Lesson

This lesson offers a pair of puzzles to enforce the skills of identifying equivalent trigonometric expressions. Additional worksheets enhance students' abilities to appreciate and use trigonometry as a tool in problem solving. This lesson is adapted from an article by Mally Moody, which appeared in the March 1992 edition of *Mathematics Teacher*.

**Prerequisites: **Students should be familiar with trigonometric functions and their definitions.

These are independent activities to be used during relevant lessons in any trigonometric unit. Each can be incorporated into a daily plan. The time required varies according to the level and ability of the students and the amount of discussion generated by the teacher and students.

**Trigonometry Practice **

For the first puzzle, Trigonometry Square Activity Sheet 1, students must apply trig values for some common angles. Prepare the activity sheet by cutting up the sheets into squares, and ideally laminating them.

Trigonometry Square Activity Sheet 1

Place the sixteen squares in an envelope. Divide students into groups (groups of three work well). Give each group an envelope of squares, and instruct them to match equivalent expressions to create one large square, lining up equivalent values. Students will review and learn common equivalents and will be better prepared to deal comfortably with trigonometric functions in more advanced studies. If a group seems to be having difficulty, suggest identifying one of the corner squares to get started. (Note that the arrangement on the original activity sheet represents one possible solution, though students may find others.)

Trigonometry Square Activity Sheet 2 is the same idea as Trigonometry Square Activity Sheet 1, except here the students will be matching equivalent expressions, using trig identities.

Trigonometry Square Activity Sheet 2

These sheets are "answered" by the correct assembly of the puzzle. Some answers will vary on these two activity sheets. Students should be encouraged to present and defend their own answers. They should also be encouraged to reflect on and accept the possibilities suggested by classmates.

**Applying Trigonometry to Situations through Angle of Elevation and Angle of Declination**

This activity includes two problems. The first uses right-triangle trigonometry to determine the height of a cliff. The second uses the law of sines. Distribute a copy of Angle of Elevation Activity Sheet to each student and allow time for individual contemplation of the material.

Angle of Elevation Activity Sheet

Encourage students to use a calculator. Students should then work in small groups, comparing ideas. As students begin work, emphasize that many "correct" answers are possible but that teams should agree on one answer to each problem. Finally, the teacher should initiate a whole-class discussion; record the consensus of answers to 1(a), 1(b), and 1(c) on the overhead transparency; and discuss the different groups' answers to problem 2.

After discussing the answers to the Angle of Elevation Sheet, distribute the Angle of Declination Activity Sheet.

Angle of Declination Activity Sheet

This activity includes one problem in which the law of cosines can be used and another that is open to a variety of approaches. The procedure for using this sheet is similar to that for the previous sheet, providing students time to devise their own problems, and then comparing results as a class.

**Sheet 1:** 1(a) Answers will vary. Some may say that Chris is standing at the base of the cliff and looking straight up; others may say that he could be back infinitely far. Others may argue that the distance would have to be at least a few feet from the base to actually identify the top. They may also say that limited ability to see will require that Chris be no more than a few miles away. 1(b) The greater the distance, the greater the height. 1(c) The greater the angle, the greater the height. 1(c) Answers will vary. Many students will simply swap angle measures. 2(b) The angle measures must be 60 degrees, but distance will vary. 2(c) The sum of the selected angle measures must be 90 degrees, but distance will vary. 2(d) One of the selected measures must be 90 degrees. Other answers will vary.

**Sheet 2** 1(a) Answers will vary. The airplane must be off the ground and high enough for the pilot to spot the villages but low enough that the villages can be discerned. 1(b) the angle measures must be between 9 degrees and 90 degrees. 1(c) The larger the angles, the closer the villages. 1(d) The greater the height, the farther apart the villages. 2. Answers will vary. Some students may elect to measure the angles adjacent to the street side of the triangle and to determine additional information by computation. Others may elect to measure the other two sides or an angle and another side.

**Assessments**

- Have students complete Trigonometry Square Activity Sheet 2 independently.
- Create a mixed set of questions including angle of elevation and angle of declination. Have students identify whether it would be angle of elevation or angle of declination for each question. Then have them solve for a missing variable.

**Extensions**

- Have students redo the Trigonometry Square Activity Sheets in a different order.
- Have students create their own Trigonometry Square Activity Sheet and exchange and solve with a partner.
- Ask students to come up with other real world scenarios in which angle of elevation and angle of declination would be applicable.

**Questions**

How did you determine the order in which you placed the squares from the Trigonometry Square Activity Sheet?

[Answers will vary]

**Reflection**

- What difficulties did students encounter while completing the Trigonometry Square Activity Sheets? How did you address them?
- How did you address any differences within the groups?
- Were there any groups that struggled considerably more than other groups? What did you do to differentiate the activity to assist with understanding?
- What could you have done to make this lesson more effective?

### Learning Objectives

Students will be able to:

- Analyze situations, check for limitations, and examine appropriate methods of solutions using trigonometry.
- Practice manipulating trigonometric functions and in substituting equivalent expressions.
- Work in small groups encouraging classmates and communicating thoughts.

### NCTM Standards and Expectations

- Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others.

- Use trigonometric relationships to determine lengths and angle measures.