Sine Rule Ambiguous Case Calculator
This sine rule ambiguous case calculator resolves the tricky SSA situation, where you know two sides and an angle that is not between them. Enter side a (opposite the known angle A), side b, and angle A in degrees, and the calculator finds the possible values of angle B using the law of sines, then tells you whether there are zero, one or two valid triangles. The ambiguity arises because the sine of an angle and the sine of its supplement are equal, so a calculated sine can match two different angles. The calculator checks each candidate against the rule that the angles of a triangle add to 180 degrees, and reports only the triangles that actually exist. The method, a worked example and the assumptions are set out below.
The formula
By the law of sines, sin B = b sin A / a. If this value is above 1 there is no triangle. Otherwise B can be the principal angle B1 = arcsin(b sin A / a) or its supplement B2 = 180 − B1. Each is valid only if A + B is less than 180 degrees, giving zero, one or two triangles.
Worked example
With a = 7, b = 10, A = 30 degrees: sin B = 10 sin 30 / 7 = 0.714286, so B1 = 45.585 degrees and B2 = 134.415 degrees. Both keep A + B under 180, so there are two triangles. Enter 7, 10, 30 to confirm.
Frequently asked questions
Why is SSA called the ambiguous case?
Because the known sine can correspond to two angles, an acute one and its obtuse supplement, so there can be two different triangles, or one, or none.
When is there no triangle?
When b sin A / a is greater than 1, the sine rule has no solution and no triangle can be formed.
When are there two triangles?
When both the acute angle B1 and its supplement B2 keep the angle sum A + B below 180 degrees.
Who this calculator is for
This calculator is for trigonometry students working through the law of sines and the SSA case.
What this calculator assumes
- Angle A is opposite side a, and you enter A in degrees.
- Sides and the angle describe a non-included (SSA) configuration.
- Results are rounded for display.
Formula and sources
Related calculators
- Trig Calculator: all six trig functions.
- Sine Calculator: sine of an angle.
- Right Angle Triangle Calculator: solve a right triangle.
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