Investigation of Cosine Law
Let's revisit the Pythagorean Theorem! 1. In the space provided below, state the Pythagorean Theorem in your own words. To add to your description, include a labelled diagram in the sketchpad window below.
2. What condition(s) must be met before this theorem can be used to calculate the length of a side of a triangle?
3. In triangle ABC, C is a right angle. What happens to the length of side c as the measure of C decreases? Using the sketchpad below, play around with the points A and B, by dragging each point, to help you answer this question.
4. Now, we will use the sketchpad below to check the Pythagorean relationship, . Is ? If not are they close? Why might they be off by a little bit?
5. What happens if you move vertex C further away from line segment AB? Use the sketchpad below to help answer your question.
6. Does the Pythagorean theorem property still apply to the triangle in question #5? Explain your reasoning.
7. Lets investigate further by moving point C, in the sketchpad window below, to 4 other locations and recording the data and calculations in the table below. We will only use acute triangles. (To complete table, copy and paste it to the answer line, then fill it out)
Triangle a b c 1 2 3 4
8. Transfer the data from the last column above to the first column below and then calculate the second column. (Again, to complete table, copy and paste it to the answer line, then fill it out)
Triangle 1 2 3 4
9. What relationship have you discovered in question #8? Think of it as the Pythagorean Theorem for Acute Triangles.
TAKE HOME MESSAGE: Generally, the cosine law can be used for any acute triangle:
Example 1
Given 2 sides and the contained angle, find the length of side a1 in the following triangle.
Example 2
Given three sides, solve for angle B in the following triangle.