Download Law of Sines

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© Boardworks 2012
The law of sines
Here is a non-right angled triangle, △ABC.
C
a is the length of the side
opposite angle A.
b
a
A
b is the length of the side
opposite angle B.
B
c
c is the length of the side
opposite angle C.
The law of sines:
a
sin A
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=
b
sin B
=
c
sin C
or
sin A
a
=
sin B
b
=
sin C
c
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Proving the law of sines
Prove the law of sines by drawing a perpendicular line
segment from any side to the opposite vertex.
h
C
h
by definition:
b
A
c
rearrange
for h:
a
hh
sin A =
h = b sin A
equating:
h = a sin B
B
D
b
sin B
if we had chosen the perpendicular
line segment from A to a, we would
have found that:
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a
b sin A = a sin B
divide both sides of the equation
by sin A ∙ sin B:
therefore:
sin B =
b
a
sin A
=
b
sin B
=
b
sin B
c
sin C

=
a
sin A
c
= sin C
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When to use the law of sines
When should we use the law of sines?
The law of sines can be used if:
C
● two angles and the length of a side
opposite one of the angles is known
b
a
B
A
c
● or if the length of two sides and the angle opposite one
C
of these sides is known.
If we do not have this information,
we must use a different method
or a different law.
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b
A
a
c
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B
Finding lengths
Use the law of sines to find the length of side a.
write the law of sines:
substitute for the given values:
multiply each side of
the equation by sin 118°:
C
evaluate:
7 cm
a
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sin A
a
sin 118°
=
=
a =
b
=
sin B
c
sin C
7
sin 39°
7 sin 118°
sin 39°
a = 9.82 in (to the nearest
hundredth)
118 °
A
a
39 °
B
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Finding angles
Use the law of sines to find m∠B.
write the law of sines:
8 in
A
46 °
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a
sin A
=
b
=
sin B
substitute for the given values:
sin B
8
=
sin 46°
6
multiply each side of
the equation by 8:
C
sin B
=
8 sin 46°
6
=
sin–1
find the inverse
for each side
6 in
of the equation:
evaluate:
B
m∠B
m∠B =
c
sin C
8 sin 46°
6
73.56° (to the
nearest hundredth)
© Boardworks 2012