# Ambiguous Case Law Of Sines

- Author:
- Gary Rubinstein

- Topic:
- Sine

## Initial Observations

Move point B around and experiment with checking in the different checkboxes and also the three sliders for angle A, side b (segment AC) and side a (segment BC). What are some observations you have about which configurations lead to 0 possible triangles, 1 triangle with angle CBA acute, 1 triangle with angle CBA right, 2 triangles -- one with CBA acute and the other with CBA obtuse, or no possible triangles.

## Question 01

If angle A is 30 and AC=10, what must be true about CB for there to only one triangle with angle CBA as a right angle?

## Question 02

If angle A is 30 and AC=10, what must be true about CB for there to be no possible triangles?

## Question 03

If angle A is 30 and AC=10, what must be true about CB for there to be two possible triangles, one with angle CBA acute and one with angle CBA obtuse?

## Question 04

If angle A is 30 and AC=10, what must be true about CB for there to only one triangle with angle CBA as an acute angle?

## Question 05

What happens when angle A is an obtuse angle. What is the range of possibilities of things that can happen when A is obtuse. Can you summarize an easy way to determine whether there will be 0, 1, or 2 triangles and whether angle CBA will be acute, right, or obtuse?

## Question 06

Revisit the initial observations question. Can you now summarize any rules that could help you determine which of the four possibilities will happen depending on angle A, side a, and side b? The four options are 1) No triangles possible, 2) One triangle possible with angle B acute, 3) One triangle possible with angle B right, 4) Two triangles possible -- one with angle B acute and the other with angle B obtuse.