Introduction: We will show and apply the Law of Sines which is a relationship among the sides and angles of a triangle. Applications will involve solving a triangle, calculating the lengths of sides and the measurements of angles, as well as calculating the area of a triangle.

The Lesson:

Given a triangle ABC with altitude AD observe that

producing the result AD = bsinC.

Also observe that

producing the result AD = csinB.

This gives us csinB = bsinC or equivalently we have .

We can extend this relationship as follows:

This relationship is known as the “Law of Sines.” It can also be written as

The Law of Sines can also be derived from a formula for the area of triangle ABC. The area of a triangle is ½ base x height.

If we use a as the base and the height AD = csinB = bsinC, then we have:

Area of triangle ABC = ½ acsinB = ½ absinC

If we draw an altitude from either B or C, then we can also state that the

Area of triangle ABC = ½ bcsinA

Summarizing,

Area of Triangle ABC = ½ acsinB = ½ absinC = ½ bcsinA.

If we multiply this equation by 2 and then divide through by abc, these area formulas become a statement of the Law of Sines.

Note: The Law of Sines involves a ratio of the sine of an angle to the length of its opposite side. Therefore it will NOT work if no angle of the triangle is known or if an angle is known but its opposite side is not.

Let's Practice:

1. In triangle ABC suppose side c = 12 and angle A = 75º and angle B = 26º. Find the length of side a.

In order to use side c in the Law of Sines, we need to know angle C.

This can be found using the fact that the three angles of ABC have a sum of 180º.

giving us

This answer is reasonable since angle A is a little smaller than angle C. We expect side a to be a little smaller than side c.

2. Suppose triangle ABC has angle A = 67º, angle B = 33º and a = 8. Find the length of side b. Also find the area of this triangle.

From the triangle, we can set up the relationships:

Area = ½ absinC .

We know that angle C is because angle C + 67º + 33º = 180º.

Thus the area of triangle ABC .

Note that in both these examples the measure of two angles is known. While the Law of Sines can be used in situations where only one angle measurement of a triangle is known, it is often more convenient to solve such a triangle by using the Law of Cosines.