Ambiguity with the Sine Law
When using the sine law to solve for the angle of a triangle,
we cannot simply rely on our calculator to provide us with the
correct answer.
example
Given triangle ABC
A=20, a=30, b=50
a/sinA = b/sinB
30/sin20 = 50/sinB
sinB = 50sin20/30
sinB = 0.570
since the sine ratio of B is positive, B could be in either quadrant
1 or 2.
B = sin(-1)0.570
B = 35, but that is not the final answer, only the reference angle
This reference angle could be in Q1 or Q2. If in Q1, the angle is
35, if in Q2, the angle is 145.
so, really,
B = 35, or 145
If B = 35, then C = 125 by SATT.
If B = 145, then C = 5 by SATT.
Both answers are valid
A = 20, a = 30, b = 10
a/sinA = b/sinB
30/sin20 = 10/sinB
sinB = 0.114
Again, B could be in quadrant 1 or 2, since the sine ratio is
positive.
B = 7
Again, this is only the reference angle. If in quadrant 1, the angle
is 7. If in qkuadrant 2, the angle is 163
B = 7, or 163
If B = 7, then C = 153 by SATT
if B = 163, then SATT is violated. Thus, 163 is not an acceptable
answer
thus, B = 7.
A = 20, a = 30, b = 100
a/sinA = b/sinB
30/sin20 = 100/sinB
sin B = 1.140
This is not possible, the sine ratio of an angle must be between 1 and
-1. Therefore, either my work is wrong (I doublechecked, I am
correct), or the triangle is not possible. Sketch the given
information to understand why it is not possible.
