Trigonometric Ratios
The Lesson Activities will help you meet these educational goals:
· Content Knowledge—You will understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
· STEM—You will apply mathematical and technology tools and knowledge to analyze real-world situations.
· 21st Century Skills—You will use critical-thinking and problem-solving skills.
Directions
You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.
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Self-Checked Activities
Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.
1. Trigonometric Ratios: Fixed Acute Angle
When designing a truss, a truss builder might know the base angle measurement and the length of the tie beam needed. The next step is to compute the height of the king post. Let’s take a look at some right triangles to see whether knowing the measure of an acute angle of a right triangle and the length of one of the sides is enough to find the lengths of the other two sides.
You will use GeoGebra to explore the trigonometric ratios when an acute angle is fixed for a given right triangle. Go to trigonometric ratios, and complete each step below. The right triangle represents half of a king post truss. If you need help, follow these instructions for using GeoGebra.
a. What is the measure of ÐBAC?
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b. How are ΔABC and ΔADE related? How do you know? Explain.
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c. Record the lengths of the sides of ΔABC and ΔADE.
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Side / Length / Side / Lengthd. Calculate the ratios in the table using the side lengths that you recorded in part c.
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Ratio / Sides / Value / Sides / Valuee. Despite the sizes of the two triangles, what do you notice about the ratios of each corresponding pair of sides in ΔABC and ΔADE?
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f. Now change the size of ΔADE by moving point D to a variety of positions. Check the Show Side Ratios box. What do you observe about the side ratios as you move point D? How do the ratios compare to the ratios that you calculated in part d?
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How did you do? Check a box below.
Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.
Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.
Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.
2. Trigonometric Ratios: Variable Acute Angle
Now you will use GeoGebra to explore the trigonometric ratios as the acute angle changes in a right triangle. Return to trigonometric ratios. Move point D until the Show Side Ratios box appears, check the box, and complete each step below.
a. Move point F to various positions so the measure of ÐA is approximately 15°, 30°, 45°, 60°, and 75°, and record the ratios of the sides of ΔABC and ΔADE.
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mÐA in ΔABC / / /mÐA in ΔADE / / /
b. What do you observe about the ratios of the side lengths of the right triangles as you change the measure of the acute angle A?
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c. What do the ratios of the side lengths for a right triangle depend on? In particular, do they depend on the lengths of the sides? What about the angle measurements of the right triangle? Use your observations from the previous exercises to explain.
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How did you do? Check a box below.
Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.
Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.
Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.
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