PPT Topic 1-Law of Sines.ppt with formulas

The Laws of Sine can be used with
Oblique triangle
Oblique triangle is a triangle that contains no
right angle.
A
B
C
a
b
c

What we already know
• The interior angles total 180.
• We can’t use the Pythagorean
Theorem. Why not?
• For later, area = ½ bh
• Larger angles are across from
longer sides and vice versa.
• The sum of two smaller sides
must be greater than the third.
A
B
C
a
b
c

There are three possible configurations that will enable
us to use the Law of Sines. They are shown below.
ASA
You may have an angle, a
side and then another angle
SAA
You may have a side and
then an angle and then
another angle
SSA
You may have two sides
and then an angle not
between them.
What this means is that you
need to already know an
angle and a side opposite it
(and one other side or angle)
to use the Law of Sines.
You don’t have an angle and side opposite it here but can easily find
the angle opposite the side you know since the sum of the angles in a
triangle must be 180°.

General Process for Law Of
Sines
1. Except for the ASA triangle, you will always
have enough information for 1 full fraction and
half of another. Start with that to find a fourth
piece of data.
2. Once you know 2 angles, you can subtract
from 180 to find the 3rd
.
3. To avoid rounding error, use given data
instead of computed data whenever possible.

Use Law of SINES when ...
• AAS - 2 angles and 1 adjacent side
• ASA - 2 angles and their included side
• ASS – (SOMETIMES) 2 sides and their adjacent angle
…you have 3 parts of a triangle and you need to find the other
3 parts.
They cannot be just ANY 3 dimensions though, or you won’t
have enough info to solve the Law of Sines equation.
Use the Law of Sines if you are given:

Draw a perpendicular line
and call the length h. We do
this so that we have a right
triangle which we already
know how to work with.
h



a
b
c
Let’s write some trig functions we know from the right
triangles formed.
c
h


sin
a
h


sin
Solve these for h
h
c 

sin h
a 

sin
Since these both = h
we can substitute

 sin
sin a
c 
divide both sides by ac
ac ac c
a

 sin
sin


The Laws of Sines
A
B
C
a
b
c
C
c
B
b
A
a
sin
sin
sin



Using the Law of Sines
Given: How do you find angle B?
a
A
B
C











c
b
a
B
m
C
m
A
m
6
.
21
7
.
16
4
.
102
b
c

Given: How do you find side b?
a
A
B
C












c
b
a
B
m
C
m
A
m
6
.
21
9
.
60
7
.
16
4
.
102
b
c

a
A
B
C












c
b
a
B
m
C
m
A
m
6
.
21
9
.
60
7
.
16
4
.
102
9
.
60
sin
4
.
102
sin
6
.
21 b

b
c

a
A
B
C
 
3
.
19
7
.
102
sin
9
.
60
sin
6
.
21
9
.
60
sin
4
.
102
sin
6
.
21



b
b
b












c
b
a
B
m
C
m
A
m
3
.
19
6
.
21
9
.
60
7
.
16
4
.
102
b
c

Given: How do you find side c?
a
A
B
C












c
b
a
B
m
C
m
A
m
3
.
19
6
.
21
9
.
60
7
.
16
4
.
102
7
.
16
sin
4
.
102
sin
6
.
21 c

b
c

Given: How do you find side c?
a
A
B
C












c
b
a
B
m
C
m
A
m
3
.
19
6
.
21
9
.
60
7
.
16
4
.
102
 
36
.
6
4
.
102
sin
7
.
16
sin
6
.
21
7
.
16
sin
4
.
102
sin
6
.
21



c
c
c
b
c

The Ambiguous Case
Look at this triangle.
If we look at where angle A
Is Acute
A B
C
b a
h
A
b
h sin



The Ambiguous Case
If we look at
If a = h, then there is one triangle
A B
C
b
a

h
A
b
h sin



The Ambiguous Case
If we look at
If a < h, then there is no triangle
A B
C
b
a
h
A
b
h sin



The Ambiguous Case
If we look at
If a > b, then there is one triangle
A B
C
b a
h
A
b
h sin



The Ambiguous Case
If we look at
If h< a <b, then there is two triangles
A B
C
b a
h
A
b
h sin


'
B

The Ambiguous Case
Do you remember the Hinge Theorem from
Geometry.
Given two sides and one angle, two different
triangles can be made.
http://mrself.weebly.com/5-5-the-hinge-theorem.html
15 15
11
11

42 
42

The Ambiguous Case
Where Angle A is Obtuse.
If a ≤ b, there is no
triangle
A
a
b

The Ambiguous Case
Where Angle A is Obtuse.
If a > b, there is one
triangle
A
a
b

Area of an Oblique triangle
Using two sides and an Angle.
SinB
ac
Area
SinC
ab
Area
SinA
bc
Area






2
1
2
1
2
1

Find the missing Angles and Sides
Given: 5
,
8
,
36 



 b
a
A

Given:







5
.
122
180
5
.
21
36
C
C
5
,
8
5
.
122
,
5
.
21
,
36 









 b
a
C
B
A

Given: 5
,
8
5
.
122
,
5
.
21
,
36 









 b
a
C
B
A
 
48
.
11
36
5
.
122
8
36
8
5
.
122







c
Sin
Sin
c
Sin
Sin
c

PPT Topic 1-Law of Sines.ppt with formulas

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