Download February 24, 2010

February 24, 2010
February 24, 2010
5.5 Law of Sines
The Law of Sines is a formula that allows you to be able to
solve oblique (non-right) triangles when given one of the
following pieces of information:
1. Two angles and an included side (ASA)
2. Two angles and a non-included side (AAS)
3. Two sides and a non-included angle (SSA)
Today you will be only looking at the first two cases,
(ASA) and (AAS).
February 24, 2010
How is the formula derived?
Oblique Triangles
C
C
b
a
a
h
A
A is acute
h
c
B
b
A
A is obtuse
c
B
February 24, 2010
Law of Sines
a
b
c
=
=
sinA
sinB
sinC
Note: sinA = 0.
or
(When does sinA = 0?)
sinA
sinB
=
a
b
=
sinC
c
February 24, 2010
To solve an oblique triangle, you need to know the measure
of at least one side and any two other parts of the triangle.
This breaks down into the following four cases.
1. Two angles and any side (AAS, ASA)
2. Two sides and an angle opposite one of them (SSA)
3. Three sides (SSS)
4. Two sides and their included angle (SAS)
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1. Find the remaining angle and sides.
C = 102.3 , B = 28.7 , b = 27.4 ft
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2. A pole tilts toward the sun at an 8 angle from
vertical, and it casts a 22 foot shadow. The angle of
elevation from the tip of the shadow to the top of the
pole is 43 . How tall is the pole?
February 24, 2010
Find the remaining angle and sides.
3. A = 50 , B = 62 , a = 4
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5.5 Day 2
Q: What methods do NOT work to prove two triangles
congruent?
Q: Explain why these methods do not work?
February 24, 2010
C
A
e
sid
side
B
February 24, 2010
In the SSA case, three possibilities can occur.
1. No such triangle exists
2. One triangles exist
3. Two triangles exist
Consider the following SSA cases and determine
whether no triangle, two triangles, or no triangle is
possible. Draw a sketch and/or explain if necessary.
A is acute
adj
opp
h
A
Condition
1. opp side < height
2. height < opp side < adj side
3. opp side > adj side
4. opp side = height
Triangles Possible
February 24, 2010
A is obtuse
1. opp side < adj side
2. opp side > adj side
Triangles Possible
February 24, 2010
In the SSA case, three possibilities can occur.
1. No such triangle exists
2. One triangles exist
3. Two triangles exist
Consider the following SSA cases:
A is acute
1.
2.
adj
adj
opp
opp
h
h
A
A
height < opp < adj
opp < height
Two
No
3.
4.
adj
adj
opp
h
h
A
A
opp = height
opp > adj
One
One
A is obtuse
2.
1.
opp
adj
A
opp
adj
A
opp < adj
No
opp > adj
One
opp
February 24, 2010
http://www.onemathematicalcat.org/Math/Geometry_obj/no_ASS_cong.htm
http://regentsprep.org/Regents/math/geometry/GP4/Ltriangles.htm
February 24, 2010
Solve the triangle.
1. a = 12 m, b = 31 m, A = 20.5
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2. a = 15cm, b = 25cm, A = 85
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3. c = 125m, b = 100m, C = 110