The Research Experience for Teachers Program /
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“Laws of Cosines and Sines”

Introduction/Motivation:

The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The law of cosines (LOC) basically generalizes the Pythagorean Theorem for all triangles while the Pythagorean Theorem only holds for right triangles. Take note, that if the angle θ is a right angle, the formula reduces to c2=a2+b2. Being able to manipulate the LOC to solve for a specific angle given only the side measures of a triangle is a useful skill in mathematics, as well as other STEM fields. Once one angle is found the law of sines (LOS) is a much more efficient way to find the other angles.

Materials List:

·  NetLogo software loaded on your computer

·  Net Logo file LOC.nlogo, available in the associated files

· 

·  Excel software

Figure 1 – NetLogo before and after Updating of Angles

Procedure:

Background:

You should have the understanding of how to manipulate the LOC; a2=b2+c2-2bccosθ, to solve for a specific angle. You should be able to enter the side lengths in an excel spreadsheet and create the formulas to calculate the specific angle and then use prior knowledge of the law of sines, sinAa=sinBb=sinCc, to calculate the other angles in order to solve the given triangle in the excel spreadsheet.

Preparation:

You need to make sure that Netlogo, and Excel or a similar spreadsheet software are available on your computer.

Lab Activity:

1.  Open Excel , and create one spreadsheet containing the following:

  1. The inputs for the three side and the calculations for the “red angle” using the LOC. Please make sure measurements are in degrees, so make sure to do the conversions in Excel as well.
  2. The calculations using LOS to calculate the blue angle. Please be specific in how you label your cells so I will know which angle is which. The yellow angle may be calculated anyway you choose.

2.  Once your spreadsheet is set up, open NetLogo and the file LOC.nlogo, available in the math folder on the student share drive, and click on the set up button to get your first triangle.

3.  Enter your data into your spreadsheet and calculate the angles, when completed, click the update angles button and record the calculated angles from NetLogo. If your calculations are off by more than 3/10 of a degree please check your calculations in Excel.

4.  Repeat steps 2 and 3 for a total of 5 triangles recording your information in your spreadsheet.

5.  Triangle exploration: When you have completed your 5 triangles. Re-click set-up button then click the go button. Once you have clicked the go button this allows you to click and grab any vertex and move it. Your objective is to try to construct a triangle where the perimeter is equal to the area. When you get it equal (or as close as you can) enter the side lengths and angle measures in excel.

Figure 2 – A sample spreadsheet (yours does NOT have to look exactly like this)

Assessment

Pre-Assessment

Students should be able to answer the following prior to completing the lab:

  1. Calculate the largest angle for the triangle with side lengths 12,35,37
  2. How would changing the angle you are looking for affect your formula in Excel?
  3. Will using the LOS always match the angle in NetLogo? Why or Why not?
  4. If using the LOS gives you an angle more than 3/10 of a degree off what should you check before you assume your calculation are wrong?

Results/Conclusions

  1. Why do you think your answers aren’t exactly the same as those calculated in NetLogo
  2. How would you quickly change your formula if I had asked you to find the “blue” angle using LOC?
  3. On the Triangle Exploration: Is there anything you can tell me about one or more of the angle measures in your triangle that has equal area and perimeter?