Math 487 Lab 6 (February 9)

This lab has several parts, some of which are with Sketchpad and some with the Lenart Sphere

Part 1 (Vector Notation and Law of Cosines)

Go to this link on Law of Cosines.

Use GSP to make a sketch like the one in the figure. Use the sketch to check your work as you answer the questions in the section The Geometry of the Law of Cosines at this link. For the moment, ignore the section called Pythagorean Theorem on the Sphere.

Checkup: After completing the section below, you should be able to anwer both of the folling questions:

• What is the cosine of the angle between (1, 0, 0) and (1, 2, 1)?
• Suppose K and P are on the unit sphere with center O, and the spherical distance between K = (0,0,1) and P = (x, 0, z) is 30 degrees. Find find x and z. Note: the spherical distance = measure of angle KOP.

Part 2 (Introduction to Lenart Spheres)

With one or two partners, open one of the Lenart Sphere boxes and check the inventory.

Drawing: You can draw on the sphere itself or you can use an overlay. You will need some water and paper towel to erase.

Great Circle Ruler:

1. Draw two great circles and measure the angle between them with the ruler along the "equator".
2. Given one great circle construct several great circles orthogonal to it. How are they related?
3. Given a great circle c and a point P, construct the great circle d through P orthogonal to c.
4. Construct the perpendicular bisector of two points using the ruler to measure distance.

Compass:

1. Construct a circle with given center through a point.
2. Given points A and B, construct C so that triangle ABC is equilateral.
3. Construct a 4-sided figure with 4 equal sides. Are the diagonals perpendicular?
4. Construct the circumcircle of a general spherical triangle as you would on the plane. What theorem must be true for this to work on the sphere?
5. Construct several concentric circles. What can you conjecture about spherical "pi"?

Part 3 (Pythagorean Theorem)

1. Construct a right triangle and measure the length of all 3 sides.
2. Go back to the earlier link on Law of Cosines. Work through the section of the Pythagorean Theorem and compare with your empirical results.