Chapter 10 - NCPN

Lesson 10.5 The Law of Sines
and Law of Cosines
Objectives
Use the Law of Sines to
solve for missing sides
and/or angle measures
in triangles.
Find the area of triangles.
The sine ratio is used to find missing side lengths in right triangles when
at least one angle measure and one side length is known. However, this
ratio cannot be used in triangles that are not right triangles.
For any triangle ABC, if a, b, and c represent the side lengths opposite
angles A, B, and C, respectively, then the Law of Sines states that
sin A
=
sin B
=
sin C .
a b c
Finding Side Lengths Using the Law of Sines
462 Chapter 10 Trigonometric Functions and Identities
The Law of Sines can be used to find missing side lengths
of triangles. It can be used to find missing side lengths in both right
triangles and non-right triangles when two of the angle measures and
at least one of the side lengths is known.
Example 1
Finding the Side Length
of a Triangle
In LMN, m∠L = 59°, m∠M = 45°, and MN = 12 yards. Find the
length of LM to the nearest tenth.
Solution
Draw and label LMN.
Find the measure of ∠N.
m∠N = 180° – 59° – 45° = 76°
Use the Law of Sines to solve for LM.
sin59° sin 76°
=
12 LM
12sin
76°
LM =
sin59°
LM ≈ 13.
6
Side LM is about 13.6 yards long.