# Deriving the Law of Sines

- Author:
- Brad Ballinger

## Behold.

## Deriving the Law of Sines

- If you're a web-browsing wizard, you might open this activity in two side-by-side views so you can keep the diagram in one view and the instructions/Q&A in the other.

## 1. Inscribed Angle Theorem

- If that's news to you, you might take this detour: https://www.geogebra.org/m/f55kbsgw.

- Use the Angle tool to measure angle ABC.
- As you move vertex B around the circle, the measure of angle ABC doesn't change until it passes A or C. At that moment, the arc intercepted by angle ABC flips from one side of AC to the other.

## 2. Angle bisector at B

- Construct the bisector of angle B like this: behind the Perpendicular Bisector tool, find the Angle Bisector tool; then click on segments a and c.
- Oh look, GeoGebra gave you the bisector you wanted and...a line perpendicular to it? Why? Well, the segments you clicked on are parts of lines, and those lines meet in an X-shape that has four angles. To bisect all four of those angles, we needed
~~four~~two bisectors. Wait...if you bisect two adjacent supplementary angles, are the bisectors always perpendicular? Hm. - OK, now move B around. The angle bisectors seem to be really interested in two special points on the circle. I mean, there are two points we haven't named yet, that the bisectors of angle B keep going through. Move B faster...maybe you can trick them.
- No? Did you try moving B past A or C?
*What is the deal with those two points?*

## Q/A #1

Strange things are happening. Why do we need only two lines to bisect four angles? Are the bisectors of a linear pair always perpendicular? Do you notice anything interesting about the two mystery points? Respond to any or all of these questions. If this is a GeoGebra Classroom activity, your teacher will be able to react to your answers. If this is NOT a GeoGebra Classroom activity, you will have given some electrons a little bit of joy before sending them into the ether.

## 3. Perpendicular bisector of the side opposite B

## The three steps above are background information.

## Q/A #2

What's the measure of angle C right now? If you're not sure, think about the Inscribed Angle Theorem again.

## Q/A #3

Now AB is a diameter of the circle and there's a right angle at C. That means ABC IS A RIGHT TRIANGLE oops, fixed my caps lock. How's your right triangle trig? I mean, what's ?

## Q/A #4

What's a pirate's favorite letter?

## Q/A #5

On the advice of Q/A #4, solve for c in the equation from Q/A #3.

## Q/A #6

As you move point B around, does side b change?

## Q/A #7

As you move point B around, does the measure of change?

- When B is positioned just right, is the diameter of the circle; and
- When you move B to other places, neither b nor sin(B) change value.

*always*the diameter of the circle. Nothing here makes B any more special than A or C. So, and are

*also*always the diameter of the circle. Since the three expressions , , and are three different ways to express the same quantity, they're all equal: . At last (drum roll, please): Equal, nonzero fractions have equal reciprocals, so , which we call the Law of Sines.