The law of sines, the law of cosines and triangle area
Solving any triangle with the law of sines and the law of cosines, the ambiguous SSA case, and the area formula using sine.
Beyond right triangles
SOH-CAH-TOA needs a right angle. For an oblique triangle (one without a right angle) we use two laws. The standard labels: sides \(a\),\(b\),\(c\) opposite angles \(A\),\(B\),\(C\).
Area with sine
If you know two sides and the included angle, the height to side \(b\) is \(a\sin C\) (it's a leg of a right triangle with hypotenuse \(a\)). So:
The law of sines
Writing the area three ways and setting them equal gives:
Use it when you know an angle and its opposite side, plus one more piece: AAS, ASA or SSA.
The ambiguous case (SSA)
With two sides and a non-included angle, there may be zero, one or two triangles. The reason: \(\sin B = k\) has two possible angles between \(0°\) and \(180°\), namely \(B\) and \(180° - B\).
The law of cosines
When \(C = 90°\),\(\cos C = 0\) and this is the Pythagorean theorem. Use the law of cosines for SAS (to find the third side) and SSS (to find an angle).
Practice
Quick check
What does the law of cosines become when the angle is \(90°\)?
\(\cos 90° = 0\), leaving \(c^2 = a^2 + b^2\).
Quick check
You know two sides and the included angle. What should you use first to find the third side?
SAS has no known angle-side pair, so the law of sines can't start.
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