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Law of Sines
Lesson 4.1
Working with Non-right Triangles

We wish to solve triangles which are
not right triangles
C
sin A  ?
sin B  ?
h
h
sin A 
sin B 
b
a
b  sin A  h  a  sin B
b  sin A  a  sin B
sin C
sin A sin B


c
a
b
b
A
a
h
c
B
View Sine Law
Spreadsheet
2
Spreadsheet Exercise

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



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Note sine ratios for all angles 60°
Change angle to 75°, record ratios
Change side a to 20, record ratios
Change side b to 5, record ratios
What is smallest sin ratio you can get?
What is largest?
Why can you change the ratio with the
angles all the same?
Check out elegance point exercises
3
Using the Sine Law

If we know two angles and one side,
we can solve the triangle

Actually, if we know two angles, we know
C = 112°
all three
B  180 - 23.5 - 112
 44.5
a
b = 216.75
A =23.5°
sin 44.5 sin 23.5 sin112


216.75
a
c
c
B
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Using the Sine Law

If we know two sides and an opposite
angle

We can solve the whole triangle
C
a =9.5
sin 47 sin A

15
9.5
b=15
A

c
Now how to find angle C and
then side c?
B = 47°
5
The Ambiguous Case (SSA)

Given two sides and an angle opposite
one of them, several possibilities exist


No solution,
side too short
to make a triangle
One solution,
side equals
altitude
1
10
20°
10
3.42
20°
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The Ambiguous Case (SSA)

Two possible triangle could result (why?)
sin A sin 20

10
5
10
5
20°
One unique solution,
the opposite side
is longer than
adjacent side
5
A
A'

Solving for A
could give either
an acute or
obtuse angle!
10
13.42
20°
7
Try It Out

Solve these triangles – watch for
ambiguous case
9.0
44
78°
28°
14
32°
8
Height of a Kite

Two observers directly under the string and
30' from each other observe a kite at an
angle of 62° and 78°. How high is the kite?
?
78°
62°
30
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Assignment
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Lesson 4.1
Page 298
Exercises 1 – 31 odd
10
Elegance Exercises
On the Sine Law spreadsheet
 Set both sides to 15





Type up your findings
in a neat format.
Report your results
with diagrams,
complete sentences.
Submit for extra points
on next test
Manipulate angle
Record at least 12 ordered pairs (angle, ratio)
Plot them, determine what kind of function this is
Determine possible function for the ordered pairs
Set the angle to 75°

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Manipulate one other side
Record at least 12 ordered pairs (side, ratio)
Plot them, determine what kind of function this is
Determine possible function for the ordered pairs
11