Oblique Triangles

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Oblique Triangles

• Law of Sines
• Law of Cosines

Law of Sines
Law of Sines
Law of Sines
Law of Sines
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by R. Arnold Moore
2
Law of Sines

• To be used when you know
• Two sides and an angle opposite one
  side (ambiguous case)
• One side and two angles

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3
The Ambiguous case
Known parts 2 sides and an angle
opposite one of known sides

• Is the given angle
• Acute
• Obtuse

4
Compare Known Sides
adjacent side
opposite side
acute angle

• Is opposite side lt, , or gt adjacent side?
• lt
• gt

5
Opposite lt Adjacent
opposite side
adjacent side
acute angle
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Since opposite side is less than the adjacent
side, draw a segment from the angle formed by the
adjacent and opposite sides. Make the segment
perpendicular to the side opposite this angle.
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From our right triangle properties, the length of
this segment is found by multiplying the adjacent
side times the sine of the acute angle.
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Adjacent times Sine of Acute Angle
adjacent side
opposite side
acute angle

• Is opposite side lt, , or gt adjacent side times
  sine of acute angle?
• lt
• gt

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Opposite gt Adjacent times Sine of Acute Angle
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adjacent side
opposite side
opposite side
acute angle
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Then there are two solutions.
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The opposite side may be located as shown
or it may be located on the other side of the
perpendicular.
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Example
END
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Opposite Adjacent times Sine of Acute Angle
adjacent side
opposite side
acute angle
If the adjacent side times the sine of the angle

opposite side
, the triangle
is a right triangle. There is exactly one
solution. Use right triangle ratios to solve.
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END
Example
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Opposite lt Adjacent times Sine of Acute Angle
opposite side
adjacent side
opposite side
acute angle
If the opposite side
lt adjacent side
times sine of the angle,
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there is no solution.
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Example
END
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Opposite gt Adjacent
adjacent side
opposite side
acute angle
If the opposite side is greater than or equal to
the adjacent side, there is one solution.
Proceed with the Law of Sines to solve the
triangle.
END
gt Example
Example
11
Obtuse Angle
opposite side
adjacent side
obtuse angle

• Is the opposite side lt , , or gt the adjacent
  side?
• lt or
• gt

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Opposite lt Adjacent
opposite side
adjacent side
obtuse angle
If the opposite side is less than or equal to the
adjacent side, there is no solution.
END
Example
13
Opposite gt Adjacent
opposite side
adjacent side
obtuse angle
If the opposite side is greater than the adjacent
side, there is one solution. Use the Law of
Sines to solve.
END
Example
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Law of Sines

• Thats that
• All done

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Example
Note Triangles are not drawn to scale
Given ÐA 40, a 75, b 85
C
You know two sides and a not-included angle, so
you will use the law of sines to solve.
75
85
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B
40
A
Since the angle is acute, compare the opposite
side (a) to the adjacent side (b).
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75 lt 85 a lt b
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Example (cont.)
Note Triangles are not drawn to scale
Given ÐA 40, a 75, b 85
C
Since the opposite is less than the adjacent, the
length of the perpendicular from ÐC to AB
(adjacent times the sine of the angle) must be
calculated.
75
85
B
40
A
D
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The opposite side (75) is greater than 54.6369
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CD 85 sin 40 54.6369
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TWO SOLUTIONS
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Example (cont.)
Note Triangles are not drawn to scale
Given ÐA 40, a 75, b 85
C
75
85
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B
40
A
D
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Since ÐB in the first solution 46.76, ÐB of
the second solution is found by subtracting
46.76 from 180
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click
click
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Example (cont.)
Note Triangles are not drawn to scale
C
C
93
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75
85
85
7
click
75
click
click
47
133
B
40
40
B
A
A
First solution drawing
Second solution drawing
click
click
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Example (cont.)
Note Triangles are not drawn to scale
Second solution drawing
C
C
First solution drawing
93
75
85
85
7
75
SOLVED
47
133
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B
40
40
B
A
A
116
14
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END
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Example
Note Triangles are not drawn to scale
Given ÐA 76.4, a 27.3, b 29.0
C
You know two sides and a not-included angle, so
you will use the law of sines to solve.
27.3
29.0
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76.4
B
A
Since the angle is acute, compare the opposite
side (a) to the adjacent side (b).
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27.3 lt 29.0 a lt b
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Example (cont.)
Note Triangles are not drawn to scale
Given ÐA 76.4, a 27.3, b 29.0
C
Since the opposite is less than the adjacent, the
length of the perpendicular from ÐC to AB
(adjacent times the sine of the angle) must be
calculated.
27.3
29.0
B
76.4
A
D
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The opposite side (27.3) is less than 28.1869.
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CD 29.0 sin 76.4 28.1869
It is too short.
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NO SOLUTION
END
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Example
Note Triangles are not drawn to scale
Given ÐB 30, b 42.6, c 85.2
A
42.6
85.2
You know two sides and a not-included angle, so
you will use the law of sines to solve.
C
30
B
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Since the angle is acute, compare the opposite
side (b) to the adjacent side (c).
42.6 lt 85.2 b lt c
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Example (cont.)
Note Triangles are not drawn to scale
Given ÐB 30, b 42.6, c 85.2
A
Since the opposite is less than the adjacent, the
length of the perpendicular from ÐC to AB
(adjacent times the sine of the angle) must be
calculated.
42.6
85.2
C
30
C
B
C
D
D
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click
The opposite side (42.6) is equal to AD. The
triangle is a right triangle with one solution.
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AD 85.2 sin 30 42.6
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Example (cont.)
Note Triangles are not drawn to scale
Given ÐB 30, b 42.6, c 85.2
A
ÐA 90 30 60 ÐC 90
SOLVED
85.2
60
42.6
90
30
B
C
73.8
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END
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Example
Note Triangles are not drawn to scale
Given ÐC 43, c 33.3, a 25.2
C
You know two sides and a not-included angle, so
you will use the law of sines to solve.
43
25.2
B
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A
33.3
Since the known angle is acute and its opposite
side is greater than the known adjacent side,
there is one solution.
Proceed with the Law of Sines to solve the
triangle.
click
click
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Example (cont)
Note Triangles are not drawn to scale
Given ÐC 43, c 33.3, a 25.2
C
43
25.2
47.0
click
click
B
31
106
SOLVED
A
33.3
ÐB 180 43 31 106
END
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Example
Note Triangles are not drawn to scale
Given ÐC 43, c 33.3, a 33.3
C
You know two sides and a not-included angle, so
you will use the law of sines to solve.
43
33.3
click
B
A
33.3
Since the known angle is acute and its opposite
side is equal to the known adjacent side, there
is one solution.
Proceed with the law of sines to solve the
triangle.
click
click
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Example (cont)
Note Triangles are not drawn to scale
Given ÐC 43, c 33.3, a 33.3
C
Since a c, then ÐA ÐC
43
33.3
48.7
SOLVED
B
43
94
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click
ÐB 180 43 43 94
A
33.3
END
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Example
Note Triangles are not drawn to scale
Given ÐB 104, b 24, a 30
You are given two sides and a not-included angle.
Therefore you will use the law of sines to solve.
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The angle is obtuse.
C
Its too short
Its too short
Its too short
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But the opposite side is less than the adjacent.
30
104
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click
A
B
NO SOLUTION
END
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Example
Note Triangles are not drawn to scale
Given ÐB 104, b 44, a 30
You are given two sides and a not-included angle.
Therefore you will use the law of sines to solve.
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The angle is obtuse.
C
The opposite side is greater than the adjacent.
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44
30
104
A
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B
There is one solution.
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Example (cont.)
Note Triangles are not drawn to scale
Given ÐB 104, b 44, a 30
C
44
35
30
click
SOLVED
104
41
A
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B
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END
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WV IGOS
T.11 solve practical problems involving
triangles using the trigonometric
functions, the Pythagorean Theorem, the
Law of Sines, and the Law of Cosines. T.19 use
a scientific calculator to solve practical
problems involving triangles.
33
Bibliography
Clemens, Stanley R., Frank Demana, Gregory D.
Foley, Bert K. Watts Precalculus, A Graphing
Approach Addison-Wesley Publishing Company,
Inc., New York, 1997
Hall, Bettye C., Jerome D. Hayden Trigonometry
Prentice Hall, Englewood Cliffs, New Jersey, 1990