Concept Byte: Exploring Trigonometric Ratios

Use With Lesson 8-3

TECHNOLOGY

Construct

Use geometry software to construct A B → eh b vector and A C → eh c vector so that ∠ A angle eh is acute. Through a point D on A B → , eh b vector , comma construct a line perpendicular to A B → eh b vector that intersects A C → eh c vector in point E.

Moving point D changes the size of Δ A D E . cap delta eh d e , . Moving point C changes the size of ∠ A . angle eh .

Exercises

1. • Measure ∠ A angle eh and find the lengths of the sides of Δ A D E . cap delta eh d e , .
   • Calculate the ratio leg opposite  ∠ A hypotenuse , fraction legopposite . angle eh , over hypotenuse end fraction . comma which is E D A E . fraction e d , over eh e end fraction . .
   • Move point D to change the size of Δ A D E cap delta eh d e without changing m ∠ A . m angle eh .

   

   What do you observe about the ratio as the size of Δ A D E cap delta eh d e changes?

2. • Move point C to change m ∠ A . m angle eh .
   1. What do you observe about the ratio as m ∠ A m angle eh changes?
   2. What value does the ratio approach as m ∠ A m angle eh approaches 0? As m ∠ A m angle eh approaches 90?
3. • Make a table that shows values for m ∠ A m angle eh and the ratio leg opposite  ∠ A hypotenuse . fraction legopposite . angle eh , over hypotenuse end fraction . . In your table, include 10, 20, 30, …, 80 for m ∠ A . m angle eh .
   • Compare your table with a table of trigonometric ratios.

   Do your values for leg opposite  ∠ A hypotenuse fraction legopposite . angle eh , over hypotenuse end fraction match the values in one of the columns of the table? What is the name of this ratio in the table?

Extend

4. Repeat Exercises 1–3 for leg adjacent to  ∠ A hypotenuse , fraction legadjacentto . angle eh , over hypotenuse end fraction . comma which is A D A E , fraction eh d , over eh e end fraction . comma and leg opposite  ∠ A leg adjacent to  ∠ A , fraction legopposite . angle eh , over legadjacentto . angle eh end fraction . comma which is E D A D . fraction e d , over eh d end fraction . .
5. • Choose a measure for ∠ A angle eh and determine the ratio r = leg opposite  ∠ A hypotenuse . r equals . fraction legopposite . angle eh , over hypotenuse end fraction . . Record m ∠ A m angle eh and this ratio.
   • Manipulate the triangle so that leg adjacent  ∠ A hypotenuse fraction legadjacent . angle eh , over hypotenuse end fraction has the same value r. Record this m ∠ A m angle eh and compare it with your first value of m ∠ A . m angle eh .
   • Repeat this procedure several times. Look for a pattern in the two measures of ∠ A angle eh that you found for different values of r.

   Make a conjecture.

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