The Law of Sines
Using The Geometer's Sketchpad

The Law of Sines is a handy triangle solving tool that's particularly useful in non-right triangles, although it works in any triangle. If the known dimensions of the triangle include one angle and the side opposite it, plus any other side or angle, the Law of Sines can be used to determine any remaining dimension of the triangle.

The Law of Sines states that

     sin A  =  sin B  =  sin C
     -----     -----     -----
       a         b         c
Open the sketch ssa.gsp. Page 1 (Intro) shows a triangle with the angles and side lengths measured. You can see that all of the Law of Sines ratios are equal. Move vertex C of the triangle to get different shapes and try to answer the following questions:
  1. How close to zero can you get the ratio of sin A/a to be?

  2. How large can the ratio be?

  3. Explain what is happening, both in the triangle and mathematically, when the ratio is at an extreme (either minimum or maximum)?

Go to page 2 of the sketchbook (Ambiguous Case). Here we are given an angle, a side, and a side, in that order. Specifically we're given angle A, side a, and side b. We can use this information in the Law of Sines to solve for angle B. Play around with a, b, and angle A and see what sort of things happen, then answer the following questions:
  1. Under what conditions is there no solution for B?

  2. Under what conditions is there only 1 solution?

  3. Under what conditions is the one solution a right angle?

  4. Under what conditions are there 2 solutions for B?

  5. Explain why there is no Side-Side-Angle Congruence Postulate.

Extra To explore the shape of the triangle when the ratio is a maximum (as we did on the first page), we can use Sketchpad's "parametric color" feature. This allows you to color an object based on a parameter, or number. In this case, we will color a circle around C based on the ratio sin A/a.

Go to the Extra page. Here a circle is attached to C. Drag C around and you'll see the circle changing color. The color of the circle is controlled by the ratio. When the ratio is bigger, the color is red. When it's smaller the color is purple. And in between itıs all the colors of the rainbow.

It's easier to see this if we can keep track of the different colors we get when we move C.

  1. Select the circle interior (not the point C) and choose "Trace Interior" from the Display menu. Drag C and see what happens.
Everything in red is where our ratio is closer to its maximum. If you want to make the red region smaller, we need to change the "range" of things that get colored red.
  1. Move C around in the red area. Make a note of the largest value you see for the ratio and the smallest value.
  2. Press ESC a couple of times to clear the traces.
  3. Choose the circle interior and the ratio sin A/a.
  4. Select Display -> Color -> Parametric....
  5. For the minimum and maximums, put in the values you found for the ratio. Make sure that "don't repeat" is the selected option (of the three coloring schemes).
  6. Click okay.
  7. Repeat steps X through Y until you are pretty sure you know what kind of triangle produces the maximum ratio!

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