USING TRIGONOMETRY
SOLVING A TRIANGLE
“Solving a triangle” means finding the sizes of all angles and sides. To get the most accurate
results, use given information whenever possible and do not approximate until the final step.
I. Given two angles and one side (ASA or AAS)
A. Find third angle by subtracting sum of known angles from .
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B. Use Law of Sines to find unknown sides.
II. Given two sides and an angle opposite one of the known sides (SSA)
A. Use Law of Sines to find angle opposite other known side. (If the sine of the angle
is greater than one, then NO triangle can be constructed with given parts.)
B. Subtract the angle found in Part A from to get second possibility. One of the
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angles found in Parts A and B will be obtuse. Add obtuse angle to given angle.
1. If resulting sum is greater than or equal to , discard the obtuse angle as
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an option. Use Law of Sines (or Law of Cosines) to find third side.
2. If resulting sum is less than , do two separate calculations as there are
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two different triangles possible. Use Law of Sines (or Law of Cosines) to
find each third side.
III. Given two sides and included angle (SAS)
A. Use Law of Cosines to find third side.
B. Use Law of Sines (or Law of Cosines) to find smaller of two unknown angles.
This angle will be opposite the shorter side. This angle will always be acute, thus
avoiding ambiguity.
C. Subtract sum of the two angles from to find third angle.
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IV. Given three sides (SSS)
A. Use Law of Cosines to find largest angle (opposite largest side). Finding this
angle first guarantees that remaining two angles will be acute; no ambiguity exists.
B. Use Law of Sines (or Law of Cosines) to find a second angle.
C. Subtract sum of the two angles from to find third angle.
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V. Given three angles (AAA) - not enough information to determine a unique triangle.
