Exploring Trigonometric Ratios

Transcription

1 Exploring Trigonometric Ratios Lesson Summary: Students will explore the trigonometric ratios by constructing a table of values and exploring the table as the shape of the triangle changes. The lab is in response to most students misunderstanding about the table of values. Key Words: Trigonometry, sine, cosine, tangent Background knowledge: Students should be familiar with basic Cabri constructions. Learning Objectives: 1) The student will use their knowledge of Cabri software to construct a right triangle consisting of rays and segments. 2) The students will label all parts of their construction and be able to manipulate the construction to show different angle measurements and side lengths. 3) The student will construct a chart showing angle measurements, side lengths and trigonometric ratios. 4) The students will explore the three trigonometric ratios and make conjectures concerning the relationships between the angle measurements and the side lengths. 5) The students will compare the chart that they have constructed to the Trigonometric Chart in their Geometry Books. Materials: Computers with Cabri Software installed or TI-92. Trigonometric Ratio Construction Worksheet Trigonometric Charts from book or hand out Writing instrument Cabri Users Guide Geometry notes concerning Trigonometric Ratios Procedure: The students are to work in pairs on the Trigonometric Ratios Construction worksheet using the Cabri Software or the TI-92. The student will follow the procedures and complete the questions in the Exploring Trigonometric Ratios Worksheet. This project should take no longer than two days. The project extension can be done individually and completed in no longer than four days after the students complete the Exploring Trigonometric Ratios project. Assessment: 1) Check to see if students are on task.

2 2) Check student knowledge by walking around and checking progress and by questioning the students on procedures. 3) Check students knowledge by accessing the worksheets handed in by each pair of students. Exploring Trigonometric Ratios Team members names: File name: _ Lab goals: Explore trigonometric ratios through the use of Cabri software or TI-92 and make conjectures about relationships between angles and side lengths. Construction: 1) Once Cabri has started, go to the Options menu and select Preferences and set the Display precisions and units to: Length = 4 (it will display lengths in 4 decimals) Angle = 1 (it will display angles in 1 decimal) Others = 4 (it will display all other unit in 4 decimals) Click apply, then OK and it should disappear. 2) Create three non-collinear points and label them A, B, and C. (Point tool) 3) Construct ray AB and ray AC. (Ray tool) 4) Create a point on ray AB between the point A and B and label it D. (Point tool) 5) Construct a perpendicular line going through point D. (Perpendicular line tool) 6) Create a point at the intersection of ray AC and the line that is perpendicular through D and label it E. (Point tool) 7) Click and hold point D and drag it to see if it enlarges and reduces the size of triangle ADE.

3 8) Click on hold point C and drag it around to check if it increases and decreases the size of angle A. 9) Construct segments AD, AE, and DE. (Segment tool) 10) Find the lengths of segments AD, AE, and DE. Type Opposite, Adjacent or Hypotenuse, followed by =, according to its relationship to <A in the right triangle in front of the measure of the segment. Drag the measurements to upper left of the workspace. (Distance and length tool) 11) Find the measure of angle EAD. Type <A= in front of the measurement. Drag the measurement to upper left of the workspace. (Angle measurement tool) 12) Align the measurements in this order in the upper right of the workspace: <A= Opposite= Adjacent= Hypotenuse= 13) Find the Sine, Cosine and Tangent of angle A. (Calculate tool) 14) Now construct a table of values. Move the cursor to an unoccupied location in the workspace to create a spreadsheet with seven columns and ten rows. (Tabulate tool) 15) Drag point C until angle A reads 5. 16) Move the cursor to the value of angle A in the upper right of the workspace. Select it and the value should appear in the table that was created. Repeat this for all the value in the upper right of the workspace. The table should now look like the one below:

4 17) Move point C until angle A is 10. Select it and the values should appear in the table that was created. Repeat this until the chart shows the values from 15 to ) Set the page setup choose landscape and organize the workspace. Print and then use the information to answer the questions. (File Menu) Questions: 1) What are the Trigonometric ratios? 2) Write the Trigonometric ratio for the Sin A. (Hint: Sin B = AC/AB) 3) Write the Trigonometric ratio for the Cos A. 4) Write the Trigonometric ratio for the Tan A. 5) As the value of angle A increases, what happens to the trigonometric ratios? 6) When Point D is moved at each of the different angle A measurements (the sides of the of the triangle increase), what happens to the trigonometric ratios?

5 7) Summarize what your group has learned about the relationship between the side length and the effect it has on all the trigonometric ratios. 8) Summarize what your group has learned about the relationship between the value of angle A and the effect it has on all the trigonometric ratios. 9) Compare the chart to the Trigonometric Ratio Chart in the book on page 688 or any Trigonometric ratio chart. Are the value the same or similar and why? Staple the print out to this packet and turn it in before the end of the period. Extension: 1) Individually, complete the same processes for Angle E, for the values 5 to 80. 2) Answer the question for the new information.

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