The ambiguous case of the sine rule is a consequence of the symmetry of the sin function:

Therefore, if a given triangle has a specific sin(A) and it is not specified whether A is acute / obtuse, then there are two possible lengths for the side opposite A:

When we type sin−1(x) into the calculator, it will always return an acute angle. To find the corresponding obtuse angle, we take

Example

In triangle ABC, the angle A=30° and the sides [BC]=10 and [AC]=12 are given. Find

• the angle B,

• the length [AB].

The angle B is opposite [AC], and A is opposite [BC], so:

This is where the symmetry of sin comes into play: B=sin−1(0.6)=36.9° or B=180°−36.9°=143°

The resulting length [AB] can be found by finding the angle C=180°−30°−(36.9° or 143°)=113° or 6.87°.

Thus sinC=0.920 or 0.120. Then