What's the Ambiguous Case of the Sine Rule?

The ambiguous case of the sine rule occurs when SSA information is given: two sides and an angle that is not between them. Because a sine value can represent two supplementary angles, the data may produce zero, one, or two valid triangles.

This is HL-only content in AHL 3.8. The mechanism is

$$\sin \theta=\sin(180^\circ-\theta),$$

so an inverse-sine result $B_1$ has the second possibility $B_2=180^\circ-B_1$. You must test whether each angle leaves a positive third angle.

Worked Example

Suppose $A=30^\circ$, $a=7$, and $b=10$, where side $a$ is opposite and side is opposite .

Using the sine rule,

$$\frac{\sin B}{10}=\frac{\sin 30^\circ}{7},$$

which gives

$$\sin B=\frac{10\sin 30^\circ}{7}=\frac{5}{7}.$$

The GDC gives the principal value $B_1=45.6^\circ$. The supplementary possibility is $B_2=180^\circ-45.6^\circ=134.4^\circ$.

For $B_1$, the third angle is $104.4^\circ$, so the triangle is valid. For $B_2$, the third angle is , so this triangle is also valid. Therefore, the same SSA data produces two triangles.

The angle-sum test is essential because inverse sine returns only one angle. The supplementary candidate is not automatically valid: if $A+B_2\geq180^\circ$, no triangle can contain those angles.

A common misconception is that the GDC's inverse-sine answer is the only solution. It is only the principal value; SSA requires checking the supplement and rejecting impossible angle sums.

Exam technique: In a calculate or determine response, write the sine-rule equation before the GDC value, show both candidates, and state which triangles are valid. Keep full precision until the final answer. Match the calculator's angle mode to the question: degree symbols mean degrees, while HL questions otherwise assume radians.