Ambiguous Case of Law of Sines

Key idea

SSA can give zero, one or two triangles.

In depth

With two sides and a non-included angle (SSA), there can be 0, 1 or 2 triangles because sin θ = sin(180° − θ). Compare the side opposite the given angle with the height b sin A.

Worked example

A = 30°, a = 4, b = 7: b sin A = 3.5 < 4 < 7, so two triangles exist.

Real-world uses

• Checking why a surveyor might get two possible triangle layouts from the same measurements

Common mistake

Giving only the acute solution without checking the obtuse one.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Trigonometry

• Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
• Phase Shift – Horizontal translation of a wave.
• Vertical Shift and Midline – Moves the wave up or down.
• General Sinusoid – Combine amplitude, period, phase and shift.
• Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
• Pythagorean Identity – The key identity from the unit circle.