Objective: To identify and evaluate sine, cosine, and tangent ratios for an acute angle of a right triangle; uses a table, calculator, or computer to find the ratio for a given angle or find the angle for a given ratio.
GA QCC: #26
Lesson: Exploration in Trigonometry
Name: ___________________________
Construct a Right Triangle:
1. Open GSP with a new sketch, select 2 points ( select the "dot" on the left menu and hold down the shift key) in which the second point to the right of the first point, use the top Pull Down menu to "construct" a segment between them, select the "arrow", and click outside the figure to "deselect." Using "the hand" along the lift menu, label the points as A and B by clicking on each point.
2. Select "the arrow," hold down the shift key, and highlight point B and the line. Use the top Pull Down menu to "construct" a perpendicular line (through point B). Then "deselect" the highlighted objects by clicking the mouse.
3.Chose "the dot" on the left menu and select a point (C) on the upper part of the perpendicular line. Chose "the arrow," hold down the shift key, select point A, construct a segment, and click to "deselect." Use the same procedure to highlight point C and point B and "construct a segment." Be sure to "deselect." Now use "the hand" to label point C and rechose the arrow when finished labeling.
4. Click on the perpendicular line below point B and, using the top Pull Down menu, chose Display and Hide Line.
5. Now, you should have right triangle ABC with right angle at B.
Calculate Ratios:
6. Chose "the hand" and click on the hypotenuse. A small letter should appear. Double click on that letter and a dialog box will appear for relabeling. Type the words, "hypotenuse" and click on OK. Using the same procedure, click on line AB to relabel as "adjacent" and line BC to relabel as "opposite." (Once any label appears, you can shift its location by dragging with the curser when it is a "hand.") Now, rechose the "arrow."
7. Measure angle CAB (by holding down shift key and selecting points C, A, B; then pull down "measure").
8. Holding down the shift key, click on segments BC ("opposite") and AC ("hypotenuse"). Then pull down "measure" and "ratio." The calculated ratio of opposite/hypotenuse should appear on the screen. In a like manner, calculate adjacent/hypotenuse, and opposite/adjacent. (Be sure to "deselect" after each operation.)
Experiment:
a. Drag point C to change the angles. When the angle changes, do the ratios change?
_____________ Why? _________________________________________________
b. Drag point A around. Notice that the shape of the triangle stays the same because of our construction. What about the ratios, do they change? ___________________
Why?_______________________________________________________________
Learn the Terminology:
9. These ratios of right triangles that you calculated have names:
opposite/hypotenuse = sine of angle A (or simply called sin A)
adjacent/hypotenuse = cosine of angle A (or simply cos A)
opposite/adjacent = tangent of angle A (or simply tan A)
and we refer to sine, cosine, and tangent as trigonometric ratios.
10. With the experimentation that you did in Experiment b., in which the angle was the same, the trigonometric ratios remained constant regardless of the size of the right triangle. (We depend on this relationship to ultimately determine the size of an angle if we can calculate one of the ratios. The angles can then be determined by using any scientific or graphing calculator.)
Practice:
a. If sin A is (opposite side length)/(hypotenuse length), then complete the ratios for
cos A = ________________________________
tan A = ________________________________
b. Drag point C so that angle A measures 30 degrees. Using the definitions above for sin A, cos A, and tan A, record the calculations for
sin 30 degrees = _______________________ (Did you get about .50?)
cos 30 degrees = _______________________ (Did you get about .87?)
tan 30 degrees = _______________________ (Did you get about .57?)
c. Without measuring, determine the measure of angle C in the right triangle. Then calculate the trigonometric ratios of that angle:
sin _____ degrees = opposite/hypotenuse = _______________________
cos _____ degrees = ________________ = _______________________
tan _____ degrees = ________________ = _______________________
Finding the Angle of a Right Triangle:
11. If we know a particular ratio of a right triangle, the angle can be determined. (This fact is very useful in solving physics type of problems.)
12. To find an angle using the GSP calculator, pull down "measure," chose "calculate," click on function and select "arcsin[" (which says you are looking for an angle whose sine ratio of known to be _____), click on the ratio of "opposite/hypotenuse" that you have already calculated on the screen, click on the close parenthesis on the calculator, and click on OK. The calculated angle should appear on the screen. It will be an angle that you already knew, but the exercise serves as a reinforcement that the procedure works.
Practice:
d. Now delete m<CAB that appears on the screen and drag point C down to shorten segment BC. Pull down the GSP calculator and, using the definition of cosine (and "arccos"), calculate the new value of angle A:
Angle A = ____________________ (Now, measure angle A.)
e. Delete m<CAB on the screen (after your check) and drag point C up to a new height. Using the definition of tangent (and "arctan"), calculate the new value of angle A:
Angle A = ____________________ (Now, measure angle A.)