Download 10.2 The Law of Sines

10.2
The Law of
Sines
Objectives:
1.
2.
Solve oblique triangles by using the Law of Sines.
Use area formulas to find areas of triangles.
The Law of Sines

Solve the triangle.
sin 115  sin 27

45
b
b sin 115   45 sin 27
45 sin 27
b
 22.5
sin 115 
C  180  115  27  38
Ex. #1
sin 115  sin 38

45
c
c sin 115   45 sin 38
45 sin 38
c
 30.6
sin 115 
Solve a Triangle
with AAS Information
For all these
triangles, side a is
called the swinging
side and side b is
called the fixed side.
Ambiguous Case:
SSA
Ambiguous Case:
SSA

Given a possible triangle ABC with b=8, c=5, and
C=54°, find angle B.
First we draw the triangle. Since the
exact shape is unknown, we start
with a baseline and estimate a 54°
angle. Side c must be across from
angle C, so we place side b = 8
adjacent to the angle and c = 5
across from it. Since the angle is
unknown, we can attach side c to b,
we just don’t know the angle in
which they meet.
Side c in this scenario is called the
swinging side.
Ex. #2
Solve a Triangle
with SSA Information

Given a possible triangle ABC with b=8, c=5, and
C=54°, find angle B.
The swinging side can form
any angle in a circular motion
attached at point A.
Depending on the height h,
from A to the baseline, the
swinging side could either
form one triangle, two
triangles, or no triangle.
Ex. #2
Solve a Triangle
with SSA Information

Given a possible triangle ABC with b=8, c=5, and
C=54°, find angle B.
To find the height h, we will use
right triangle trigonometry:
h
sin 54 
8
h  8 sin 54
h  6. 5
The height is longer than the
swinging side, which cannot reach
the baseline to form a triangle.
Ex. #2
Solve a Triangle
with SSA Information

Lighthouse B is 3 miles east of lighthouse A. Boat C
leaves lighthouse B and sails in a straight line. At
the moment that the boat is 5 miles from lighthouse
B, an observer at lighthouse A notes that the angle
determined by the boat, lighthouse A (the vertex),
and lighthouse B is 65°. Approximately how far is
the boat from lighthouse A at that moment?
In this situation, the swinging side is now a = 5 as angle
A is given. Because the swinging side is longer than
the fixed side c = 3, it will automatically be longer than
the height h and form exactly one triangle.
h
Ex. #3
Solve a Triangle
with SSA Information

Approximately how far is the boat from lighthouse A
at that moment?
To find side b, we must first find angle B. In
order to find angle B we must first find angle C.
sin 65 sin C 

5
3
3 sin 65  5 sin C 
3 sin 65
sin C  
5
 3 sin 65 
C  sin 1

5


C  32.94
Ex. #3
Solve a Triangle
with SSA Information

Approximately how far is the boat from lighthouse A
at that moment?
B  180  65  32.94  82.06
sin 82.06 sin 65

b
5
b sin 65  5 sin 82.06
5 sin 82.06
b
sin 65
b  5.5 miles
Ex. #3
Solve a Triangle
with SSA Information

Solve triangle ABC when a = 4.8, b = 6, & A = 28°.
Here we must again first find the height in order to see if the swinging
side a = 4.8 will reach down and touch the baseline to form a triangle.
h
sin 28 
6
h  6 sin 28
h  2.8
Ex. #4
Solve a Triangle
with SSA Information

Solve triangle ABC when a = 4.8, b = 6, & A = 28°.
With a height of 2.8, the swinging side is long enough to touch the
baseline, but instead of forming just one triangle, it forms two triangles.
2.8
This occurs whenever the swinging side is longer
than the height but shorter than the fixed side.
Ex. #4
Solve a Triangle
with SSA Information

Solve triangle ABC when a = 4.8, b = 6, & A = 28°.
Since two possible triangles
are formed, both need solved.
The small numbers next to the
sides and angles indicate
which triangle we are solving.
sin 28 sin B1

4.8
6
4.8 sin B1  6 sin 28
6 sin 28
4.8
 6 sin 28 
B1  sin 1

 4.8 
B1  35.93
sin B1 
Ex. #4
C1  180  28  35.93  116.07
sin 28 sin 116.07

4.8
c1
c1 sin 28  4.8 sin 116.07
4.8 sin 116.07
sin 28
c1  9.2
c1 
Solve a Triangle
with SSA Information

Solve triangle ABC when a = 4.8, b = 6, & A = 28°.
To solve the second triangle, we must
first use a little geometry and our
answer for angle B1 to find angle B2.
Since both legs (formed by the swinging
side) are the same length, they form an
isosceles triangle. This means the two
interior angles are congruent as well. Since
we found B1 to be 35.93°, to find B2 we
subtract it from 180°.
B2  180  35.93  144.07
Ex. #4
Solve a Triangle
with SSA Information

Solve triangle ABC when a = 4.8, b = 6, & A = 28°.
To find angle C2 we simplify must
subtract angles A and B2 from 180°.
C2  180  28  144.07  7.93
Finally we can solve for side c2.
sin 28 sin 7.93

4.8
c2
c2 sin 28  4.8 sin 7.93
4.8 sin 7.93
sin 28
c2  1.4
c2 
Ex. #4
Triangle 1 :
B1  35.93
Triangle 2 :
B2  144 .07
C1  116 .07
C2  7.93
c1  9.2
c2  1.4
Solve a Triangle
with SSA Information

Two surveyors, standing at points A and B, are
measuring a building. The surveyor at point A is 20
feet further away from the building than the
surveyor at point B. The angle of elevation of the
top of the building from point A is 58°, and the
angle of elevation of the top of the building from
point B is 72°. How tall is the building?
In order to figure out the height of the
building, we will first need to find side a.
Ex. #5
Solve a Triangle
with ASA Information

How tall is the building?
To find side a, we will need the
other angles in its triangle.
180  72  108
180  58  108  14
14°
108°
Ex. #5
sin 14 sin 58

20
a
a sin 14  20 sin 58
20 sin 58
a
sin 14
a  70.11
Solve a Triangle
with ASA Information

How tall is the building?
h
sin 72 
a
14°
108°
Ex. #5
To find the height of the building h,
we will use simple right triangle
trigonometry.
h
sin 72 
70.11
h  70.11sin 72
h  66.7 ft
Solve a Triangle
with ASA Information
Caution! This formula can only be directly used when
given SAS information.
Area of a Triangle Given SAS

Find the area of the triangle shown in the figure below:
1
A  ab sin C
2
1
 56 sin 153 
2
 15 sin 153 
 6.8 cm2
Ex. #6
Find Area with
SAS Information
Heron’s Formula:
Use this formula when all sides of a triangle are known.
The variable s represents the semi-perimeter which is half
the perimeter of the triangle.
Area of a Triangle Given SSS

Find the area of the triangle whose sides have
lengths of 8, 10, and 14.
First find the
semi-perimeter:
s
1
a  b  c 
2
1
 8  10  14 
2
1
 32   16
2
Ex. #7
Then plug into Heron’s
formula and simplify:
A  ss  a s  b s  c 
 1616  816  10 16  14 
 1686 2 
 1536
 39.2 square units
Find Area with
SSS Information