This week in PC11, we learned about the ambiguous case of the sine law. At first, I was really confused on how to check it and I wasn’t really sure on how it makes sense. But now I understand why it is possible to have 2 triangles.

For example:

angle A = 40°

side a = 10

side b = 12

Angle B = x

The first thing that I need to do is find angle B using the sine law:

sin(B) / 12 = sin(40°) / 10
sin(B) = (12 × sin(40°)) / 10
sin(B) = 0.771
B = 50.4°

Now I need the check the angle in the second quadrant:

180° – 50.4° = 129.6°

Then I checked:

40° + 129.6° = 169.6°, which is less than 180°, so a second triangle is possible.

I always have to check if a second triangle is possible by subtracting the angle from 180°. It’s a little more work, but it makes sense.