Lesson Worksheet 1
To do this part of the lesson, download this Geometer's Sketchpad file.
Open the Sketchpad file and follow the following steps.
The first thing we need to is to make the angle the ladder forms with the ground to 75 degrees.
STEP 1: MEASURE < BCA (SELECT the points B, C and A in order and go to the MEASURE menu and select ANGLE. Write down your measurement ______.
STEP 2: SELECT and MOVE point C so that the angle you just measured is 75 degrees.
We now know the angle measure and one side (the height of the building). Using trigonometry we can now find every angle and side in the triangle.
STEP 3: Double-click the SHOW button on the top of the screen. This will tell you the LENGTH of the LADDER for a 75 degree angle. Write the answer here. ______.
STEP 4: Select BOTH the measurements (the ladder and the building) and go the menu MEASURE and select CALCULATE....
STEP 5: Under VALUES select the length of the building and then click / (for divide) and finally under values select the length of ladder . Click OK. Write your answer here. ______.
STEP 6: Next, select the angle measurement you found in STEP 1. Under the MEASURE menu, select CALCULATE.....
STEP 7: Under FUNCTIONS select the option SIN( and then under VALUES select < BCA . Click OK. Write your answer here. ______.
STEP 8: What do you notice about the ration you calculated and the SIN of the angle? ______.
It is common to label the sides of a triangle as in the figure below.
NOTE that the side names are dependent on the angle you are considering except for the HYPOTENUSE which is always the side across from the right angle.
STEP 9: Write a ratio using the names of the sides in the picture above to describe the SIN of < BAC.
SIN of < BCA = ______
Lesson Worksheet 2
To do this part of the lesson, download this Geometer's Sketchpad file and open it.
Recall from the previous section that you found the SIN of an angle to be the ratio of the OPPOSITE side to the HYPOTENUSE. As you may have thought there are ratios that can be formed using the other sides of the triangle. In this section, we will investigate the other major types of trig ratios.
In the Sketchpad file we have three circles all with radii of 1. Each is called the UNIT CIRCLE. We will use it to investigate the other trig ratios.
STEP 1: Consider this diagram.
Look at triangle NIS. The HYPOTENUSE is the same as the radius of the circle. Therefore, it has a measure of 1. Recall the SIN ratio we discovered in the last section for angle SIX. It is,
SIN (< XIS) = OPPOSITE (SN) / HYPOTENUSE (IS)
Since the HYPOTENUSE = 1, the SIN (angle XIS) = the length of the OPPOSITE side. That's what makes the UNIT CIRCLE special!
STEP 2:MEASURE <SIX ______.
MEASURE the SIN of <SIX ______.
Now, measure the length of SN? ______.
What do you notice about the measures? ______.
Explain why this is so.
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STEP 3:Double-click the ANIMATE button under the SIN circle.
Form a conjuncture about the relationship between the side length (SN) and the SIN of the angle.
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For the next steps use the circle under the COS label.
STEP 4: MEASURE <XOC ______.
MEASURE the cosine (COS) of <XOC ______.
Now, measure the length of OS? ______.
What do you notice about the measures? ______.
STEP 5:Double-click the ANIMATE button under the COS circle.
Form a conjuncture about the relationship between the side length (OS) and the COS of the angle.
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STEP 6: Write the ratio for the COS <XOC ______.
STEP 7: Use the figure below to write the ratio for the COS <BCA ______.
For the next steps use the circle under the TAN label.
STEP 8: MEASURE <TAX ______.
MEASURE the Tangent (TAN) of <TAX ______.
Now, measure the length of TA, TN, and AN? ______.
Do you notice anything about the measures of the sides and the TAN <TAX? ______.
CALCULATE the ratio of SIDE (TN) to SIDE (AN) ______.
Now, do you notice anything about the measures of the sides and the TAN <TAX? ______.
STEP 9:Double-click the ANIMATE button under the TAN circle.
Form a conjucture about the realtionship between the side lengths (TN) and side (AN) and the TAN of the angle.
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STEP 10: Write the ratio for the TAN <TAX ______.
STEP 11: Use the figure below to write the ratio for the TAN <BCA ______.
Lesson Worksheet 3
To do this part of the lesson, download this Geometer's Sketchpad file.
Open the Sketchpad file and do the following steps.
In this section, we will get more familiar with how to use the UNIT CIRCLE to further your understanding of trig ratios.
STEP 1: Double-click the ANIMATE button. Let the animation finish and DO NOT do anything to the sketch.
STEP 2: Notice the top circle and specifically the red segment of the triangle inside the circle. The length of this red line is also the SIN of <NIS as we saw in the previous section.
STEP 3: Go to the EDIT menu and select UNDO Translate points.
STEP 4: As the radius of the circle rotates around the circle, < SIN increases starting from 0. Double-click the ANIMATE button again. The point that is being traced shows the length of the red segment as the angle increases from 0 to 360 degrees. Describe the graph that is traced out. ______
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STEP 5: What's the length of the segment (or the SIN of the angle) at 0 degrees ______, 90 degrees ______,180 ______, and 270 ______.
STEP 6: Go to the EDIT menu and select UNDO Translate points.
STEP 7: Double-click the ANIMATE button again. Now, notice the bottom circle and the red segment of the triangle inside the circle. The length of this red line is also the COS of <SOC as we saw in the previous section.
STEP 8: Go to the EDIT menu and select UNDO Translate points.
STEP 9: As the radius of the circle rotates around the circle, < SOC increases starting from 0. Double-click the ANIMATE button again. The point that is being traced shows the length of the red segment as the angle increases from 0 to 360 degrees. Describe the graph that is traced out.
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STEP 10: What's the length of the segment (or the COS of the angle) at 0 degrees ______, 90 degrees ______,180 ______, and 270 ______.
STEP 11: Describe both the SIN and the COS graphs. What's similar and what's different between the size and shape of the graphs.
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