Law of Sines

Written by tutor Carol B.

The law of sines is a proportion used to solve for unknown sides and/or angles of any triangle. In any triangle, the ratio of a side length to the sine of its opposite angle is the same for all three sides.

Law of sines formula:

Triangle with sides a, b, c, and corresponding angles A, B, and C

a/sin A = b/sin B = c/sin C

Use the law of sines if (1) one side length and its opposite angle measurement are known, AND (2) one or more other sides or angle measurements are known. If only two sides and an included angle of a triangle are known (SAS), or if only the three side lengths of a triangle are known (SSS), we cannot use the law of sines since we cannot set up any known proportions.

AAS example:

To find the length of side a we use the law of sines and the information given.

Law of Sines via AAS

a/sin A = c/sin C
a/sin (55°) = 8/sin (30°)
solving for a gives
a = 8sin(55°)/sin(30°)
a ≅ 8(.82)/.5 ≅ 13.12

Note that when any two angles are known, the third angle is easily found since the interior angles of a triangle always sum to 180°. Therefore, anytime at least two interior angles are known and one side length is known, the law of sines can be used to find the other side lengths.

SSA situation and the Ambiguous Case:

When the lengths of two sides and the measure of a nonincluded angle are known, the law of sines can be used to solve for the unknowns. However, there are three possible solutions:

1. No solution
2. Two different solutions
3. Exactly one solution

Consider a triangle in which a, b, and A are known.

There is no solution if A is acute and a < h="" (where="" h="" is="" the="" altitude="" from="" vertex="" c="" to="" side="" c)="" or="" if="" a="" is="" obtuse="" and="" a="" ≤="">

Law of Sines via SSA

If A is acute and h < a="">< b,="" then="" there="" are="">two possible solutions.

Law of Sines via SSA - acute angles

For every other case, there is exactly one solution.

Law of Sines via SSA - A is acute

Law of Sines via SSA - A is obtuse

Law of sines practice quiz

Given triangle ABC with A = 35°, B = 25°, and a = 15 m, what is the length, to the nearest meter, of side c?

A. 11
B. 23
C. 26
D. 31
E. 46
The correct answer here would be B.

True or False:
Given triangle ABC as shown, the law of sines can be used to find the angle measurement of < c="" and="">< b.="">

A. True
B. False
The correct answer here would be B.
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