SOH-CAH-TOA
Teacher
SOH-CAH-TOA
Understanding trigonometric ratios
Description
An exploratory exercise to determine relations of equilateral, 30o-60o-90o, and 45o-45o-90o triangles.
Also, a good understanding of sine, cosine, and tangent will be formed by the use of the Geolegs and some extra activities.
Objectives (Lessons Learned)
· Understand the laws of equilateral, 30o-60o-90o, and 45o-45o-90o triangles.
· Understand how sin, cos, and tan can be calculated without a calculator.
· Understand the similarity between the calculator answer to sin(x) and opposite/hypotenuse.
· Feel comfortable with the plots of a sin, cos, and tangent function.
Sunshine State Standards/Benchmarks
· MA.C.6.4.1 represents and applies geometric properties and relationships to solve real world and mathematical problems including ratio, proportion, and properties of right triangles (PYTHAG. THM)
Bodies of Knowledge (Approved September 2007)
· MA.912.T.1.1 Convert between degree and radian measures
· MA.912.T.1.2 Define and determine sine and cosine using the unit circle
· MA.912.T.1.3 State and use exact values of trigonometric functions for special angels (degree and radian measures)
· MA.912.T.1.4 Find approximate values of trigonometric and inverse trigonometric functions using appropriate technology
· MA.912.T.1.5 Make connections between right triangle ratios, trigonometric functions, and circular functions
Relevance
Geometry and trigonometry plays an important role in everyday life. Engineers can use the theorems of geometry to solve complicated Turbomachinery problems and Chemists can find out bond energy by using angles. A good understanding of geometry and trigonometry should be acquired in order to be a successful problem solver.
Learning Challenges
Inquiry Questions
1. What kinds of side/angle relationships do you notice by observing the created triangles?
2. Does there seem to be any type of similarities between the triangles?
3. Why do you think the plot of opposite/hypotenuse looks like it does?
Conclusion Statement
The sides of a 30-60-90 triangle have a relationship that is x, 2x, and x*sqrt(3). The sides of a 45-45-90 triangle have a relationship that is x, x, and x*sqrt(2). Sin = opposite/hypotenuse, cos = adjacent/hypotenuse, and tan opposite/adjacent.
The “Aha!” Moments
1. When the student realizes that, “hey, the reason why 30-60-90 triangles have the relationship they do is simply from dividing an equilateral triangle in half”.
2. When the student realizes that, “hey, the reason why 45-45-90 triangles have the relationship they do is simply from dividing a square in half”.
3. When the student sees that the formulas they are so familiar with (SohCahToa) are actually equal to the algebraic operation of sin(x), cos(x), and tan(x) on the calculators and why using the mini triangles created from the plates.
Tools Needed
· Geolegs
· Pencil/Paper/Markers
· Calculator
· Thin white paper plates
· Straight edges
Inquiry Procedure/Assignment
30-60-90 Triangle
1. Instruct students to create two equilateral triangles using the supplies given. You may want to suggest they put their length measurements on the sides of the triangles. You may want to assign some students to measure in inches and others to measure in centimeters or millimeters.
2. Do you notice any relationship between the side lengths and the angles? If the sides of the triangle were increased, would the angles increase? Would they Decrease? Why?
3. Cut out the triangles and compare to other triangles around the room. What can be said about these triangles?
4. How can we make a right triangle out of the triangles we have? Discuss.
5. What is different about the angles in the new smaller triangle than the equilateral triangle? What is different about the sides? What are the measurements of the angles in the new right triangle?
6. On one of your new right triangles label the hypotenuse and determine the length of any missing sides. Compare to other triangles around the room. Do you see any similarities?
7. Let’s pretend we don’t know the length of the hypotenuse of the right triangle and label it “x”. How can you determine the “length” of the other sides of the right triangle? Hint: remember this triangle came from your equilateral triangle.
8. Compare your answers to the other triangles in the class. What do you see similar? What do you see that is different?
9. Will the relationship of side lengths in terms of a variable (x in our case) hold true if the length of the sides change but the angles stay the same? Why?
45-45-90 Triangle
1. Instruct students to create two squares using the supplies given. You may want to suggest they put their length measurements on the sides of the square. You may want to assign some students to measure in inches and others to measure in centimeters or millimeters.
2. How can we create two right triangles from the squares? Do you notice any relationship between the side lengths and the angles?
3. On one of your new right triangles label the hypotenuse and determine the length of any missing sides. Do you see any similarities? What do you think the angles measures are? Compare to other triangles around the room.
4. What is different about the angles in the new triangle than the square? What is different about the sides? What are the measurements of the angles in the new right triangle?
5. Let’s pretend we don’t know the length of the hypotenuse of the right triangle and label it “x”. How can you determine the “length” of the other sides of the right triangle? Hint: remember this triangle came from your square.
6. Compare your answers to the other triangles in the class. What do you see similar? What do you see that is different?
7. Will the relationship of side lengths in terms of a variable (x in our case) hold true if the length of the sides change but the angles stay the same? Why?
Unit Circle
1. Using the supplies provided make an x and y axis on your paper plate. Label them clearly on your plate.
2. Determine a method to show a 45 degree angle in quadrant one. Discuss your method with another classmate and draw your 45 degree angle on the plate.
3. How can you show 45 degree angles in each quadrant? Discuss your method with another classmate and draw 45 degree angles in each quadrant. We are measuring these angles relative to the x-axis, not the y-axis.
4. Determine a method to show a 30 degree angle in quadrant one. Discuss your method with another classmate and draw your 30 degree angle on the plate.
5. How can you show 30 degree angles in each quadrant? Discuss your method with another classmate and draw 30 degree angles in each quadrant. We are measuring these angles relative to the x-axis, not the y-axis.
6. Determine a method to show a 60 degree angle in quadrant one. Discuss your method with another classmate and draw your 60 degree angle on the plate.
7. How can you show 60 degree angles in each quadrant? Discuss your method with another classmate and draw 60 degree angles in each quadrant. We are measuring these angles relative to the x-axis, not the y-axis.
8. How do your triangles from the previous activities fit on this circle? Can you create right triangles on the circle from your 30-60-90 and 45-45-90 triangles? Practice with a classmate and show the class. Teacher circulates to be sure the triangle is in the correct position.
9. Given the following information, using your triangle measures and your calculator, complete the table
/ sin( /0o
30o /
45o
60o
90o
120o
135o
150o
180o
210o
225o
240o
270o
300o
315o
330o
360o
Sine Wave
1. With your partner, draw an x and y axis on a piece of graph paper.
2. How can we take the information from the table above and transfer this information to the x and y axis you just drew? Is it realistic to draw degrees on the coordinate system? What measure would be better? Teacher lead discussion to convert the degrees to radians measure.
3. What do you see developing on the x and y axis?
4. Extension: Have students create a table for cosine and tangent. Discuss similarities and differences to the sine table. If time allows, students can draw the cosine and tangent curves.
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Geometry/Trigonometry