The ambiguous case of the sine rule means that there are two possible solutions to the triangle.  This can only occur when we are given two sides and a non-included acute angle.  Furthermore, if the acute angle is opposite the smaller of the two given sides, then we will have two possible solutions.

The reason for this is that there are two angles, one acute and one obtuse for which the value of the sine of these angles will be the same.  Recall that sine is positive in the first and second quadrants, so under certain conditions, it is possible to have both acute and obtuse angles satisfy the problem. (See Trigonometric Ratios for Large Angles)

Solution of this type of problem is no different apart form the extra step involved.  Once the second angle is determined, you must subtract it from 180° to find the second possibility.  You will then have enough information to determine both solutions.  This is really a double-whammy problem for students!

Slide rule diagrams or instructions will not be given in this example.

Example: Solve for the triangle given the following information: A=30.3°, a=15.4, b=20.8

This case is said to be ambiguous:

1. We have two sides and a non-included angle.
2. The acute angle is opposite the smaller of the two given side.

Using the sine rule to determine angle B°.

we find that B°=43° and so C= 180°-(43°+30.3°) =106.7°.  To find the second possible angle for B°, we subtract 43° from 180° giving 137°.  So our second possible angle for C° is 180°-(137°+30.3°)=12.7°.

We now have two cases:

1. A°=30.3°, B°=43°, C°=106.7°a=15.4, b=20.8
2. A°=30.3°, B°=137°, C°=12.7°, a=15.4, b=20.8

For case 1:

giving c=29.2

For case 2:

giving c=6.71

 

Note:

1. Some people just prefer to calculate the other angle anyway and see if it gives a valid solution.  I think it's nice to know what to expect!
2. In come cases setting the initial side using the CF scale instead of the C scale makes it easier to read off multiple angle values from the S scale.