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Your Opposite is My Adjacent

Module by: Interactive Mathematics Program

Intent

This activity follows the introduction of the trigonometric functions with an investigation into the relationship between sine and cosine.

Mathematics

The sine and cosine are cofunctions, short for complementary functions. These functions are defined at this point as ratios of sides in a right triangle. In any right triangle, the side opposite one acute angle is adjacent to the other acute angle, and these two angles are complementary angles. If the two acute angles are A and B, then the side opposite angle A is adjacent to angle B, and sin A, which is the ratio angle opposite ÐAhypotenuseangle opposite ÐAhypotenuse size 12{ { {"angle opposite "ÐA} over {"hypotenuse"} } } {}, is equal to cos B, which is the ratio. Because angle A and angle B are complementary, B = 90º – A and sinA = cos(90º – A).

Figure 1
Figure 1 (graphics1.jpg)

Progression

Students work on the activity individually and share results in their groups and with the class.

Approximate Time

15 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, then groups, followed by whole-class discussion

Doing the Activity

This activity requires little or no introduction.

Discussing and Debriefing the Activity

Have students share results in their groups. You might ask each group to be ready to answer one of the questions.

For Question 1, students should see that A = B = 90º or, equivalently, A = 90º + B or B = 90º – A. For the discussion of Question 3, it will be helpful if they see this relationship stated in all three ways. You might bring out the equations A = 90º – B and B = 90º – A by asking, How could you get one angle if you knew the other?

Review the term complementary angles (first introduced in the discussion of Very Special Triangles) for a pair of angles with a sum of 90°.

For Question 2, students should recognize that this ratio is both sin A and cos B. You can take this opportunity to point out that the word cosine begins with “co,” as does the word complementary.

For Question 3, students should be able to put Questions 1 and 2 together to get the general formulas

sin A = cos(90º – A)cos A = cos(90º – B)

Key Question

How could you get one angle if you knew the other?

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