# 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:

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.

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=”” ≤=””>

If A is acute and h < a=””>< b,=”” then=”” there=”” are=””>__two possible solutions__.

For every other case, there is exactly one solution.

## 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

**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

**B**.