## Trigonometry for Solving Problems

- Lesson

This lesson offers a pair of puzzles to enforce the skills of
identifying equivalent trigonometric expressons. Addtional 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 1, students must apply trig values for some common angles. Prepare the activity sheet but cutting up the sheets into squares, and ideally laminating them.

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

Activity Sheet: Trigonometry Square 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**

**Angle of Elevation**
This activity includes two problems. The first uses right-triangle
trigonometry to determine the heigth of a cliff.
The second uses the law of sines. Distribute a copy of Sheet 1 to each
student and allow time for individual contemplation of the material.

Activity Sheet: Angle of Elevation |

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 of problems. 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.
*Answers*: 1(a), 1(b), 1(c), 2(b), 2(c), and 2(d).

**Angle of Declination**
After discussing the answers to the Angle of Elevation Sheet, distribute the next one.

Activity Sheet: Angle of Declination |

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
actually to 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.

- Trigonometry Square Activity Sheet 1
- Trigonometry Square Activity Sheet 2
- Activity Sheet 1
- Activity Sheet 2
- Calculators
- Transparencies

### 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