Download Notes 5.6

Section 5.6
The Law of Sines
Objective: Students will be able to:
1. Solve triangles using the Law of Sines if the measures of
2 angles and a side are given.
2. Find the area of a triangle if the measures of 2 sides and
the included angle or the measures of 2 angles and a
side are given.
Law of Sines
The ___________________
can be used to solve triangles that are not right triangles.
Law of Sines
Let ∆𝐴𝐵𝐶 be any triangle with a, b and c representing the measures of the sides opposite
the angles with measures A, B, and C respectively. Then, the following is true:
𝑎
𝑏
𝑐
=
=
sin 𝐴
sin 𝐵
sin 𝐶
From geometry, you know that a unique triangle can be formed if you know the measures
of 2 angles and the included side (ASA) or the measures of 2 angles and the non-included
side (AAS). Therefore, there is one unique solution when you use the Law of Sines to solve
a triangle given the measures of 2 angles and one side.
*** In Section 5.7, you will learn how to use the Law of Sines when the measures of 2 sides
and a nonincluded angle are given.
Example 1:
Solve ABC if A = 24°, B = 62°, and a = 21.4.
First, find the measure of C.
C = 180° - (24° + 62°)
or 94°
Use the Law of Sines to find b and c.
Find b :
𝑎
sin 𝐴
𝑏
= sin 𝐵
Find c :
𝑐
sin 𝐶
=
𝑎
sin 𝐴
21.4
𝑏
=
sin 24° sin 62°
𝑐
21.4
=
sin 94° sin 24°
21.4 sin 62°
=𝑏
sin 24°
𝑐=
46.45531404
𝑏 ≈ 46.5
b
21.4 sin 94°
sin 24°
52.48573257
𝑐 ≈ 52.5
c
Example 2: The angle of depression from a window of a house to the front edge of the
swimming pool is 26.6°. The angle of depression from this same window to
the back edge of the swimming pool is 15.3°. The length of the pool is 25
feet. If a person looks out the window, about how far is he from the front
edge of the pool?
*** Make a diagram of the problem. Remember that the angle of elevation is congruent to
the angle of depression because they are alternate interior angles.
First, find .
 = 26.6° - 15.3° or
11.3°
Use the Law of Sines to find d.
𝑑
25
=
sin 15.3°
sin 11.3°
25 sin 15.3°
𝑑=
sin 11.3°
d
33.66652748
The person would be about 33.7 feet from the
front edge of the pool.
The area of any triangle can be expressed in terms of 2 sides of a triangle and the
measure of the included angle.
Area of Triangles
Let ∆ 𝐴𝐵𝐶 be any triangle with a, b, and c representing the measures of the sides
opposite the angles with measurements A, B, and C, respectively. Then the area K can
be determined using one of the following formulas:
𝐾=
1
𝑏𝑐 sin 𝐴
2
,
𝐾=
1
𝑎𝑐 sin 𝐵
2
,
𝐾=
Example 3:
Find the area of ABC if b = 14.8, c = 10.2, and A = 54°12.
1
K = 𝑏𝑐 sin 𝐴
2
1
2
𝐾 = (14.8)(10.2) sin (54° 12′ )
K
61.21909706
The area of ABC is about 61.2 square units.
1
𝑎𝑏 sin 𝐶
2
You can also find the area of a triangle if you know the measures of one side and 2
𝑏
𝑐
𝑐 sin 𝐵
angles of the triangle. By the Laws of Sines , sin 𝐵 = sin 𝐶 𝑜𝑟 𝑏 = sin 𝐶 , if you
substitute
𝑐 sin 𝐵
sin 𝐶
1
𝟏
for b in K = 2 𝑏𝑐 sin 𝐴, the result is K = 𝟐 𝒄𝟐
𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩
𝐬𝐢𝐧 𝑪
.
Two similar formulas can be developed from the Laws of Sines:
𝟏
𝑲 = 𝟐 𝒂𝟐
𝐬𝐢𝐧 𝑩 𝐬𝐢𝐧 𝑪
𝐬𝐢𝐧 𝑨
and
𝟏
𝑲 = 𝟐 𝒃𝟐
𝒔𝒊𝒏 𝑨 𝒔𝒊𝒏 𝑪
𝒔𝒊𝒏 𝑩
Example 4:
Find the area of DEF if e = 18.6, E = 78.2°, and F = 41.3°.
First find the measure of D.
D = 180° - (78.2° + 41.3°)
or
Then, find the area of the triangle.
1 sin 𝐷 sin 𝐹
K = 𝑒2
2
sin 𝐸
1
sin 60.5° sin 41.3°
𝐾 = (18.6)2
2
sin 78.2°
60.5°
K
101.5111719
The area of DEF is about 101.5 square units.