Chapter 9

Chapter 9 -...

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Unformatted text preview: Chapter
9.1:


The
law
of
sines
and
cosines
 
 In
this
section
we
will
discuss
two
laws,
the
law
of
sines
and
cosines
that
establishes
 relationship
between
the
sides
and
angels
in
any
triangle
(acute,
right
or
obtuse
 triangles).

The
purpose
of
these
laws
is
to
solve
a
triangle
(find
out
the
lengths
of
 the
three
sides
and
the
measurements
of
the
three
angels)
when
you
know
only
 some
of
the
side
lengths
or
some
of
the
angel
measurements.

You
will
have
to
 decide
which
of
the
laws
is
better
to
use
given
the
provided
information
in
the
 question.


 
 
 The
Law
of
sines:

In
any
triangle,
the
ratio
of
the
sine
of
an
angle
to
the
lengths
of
 the
opposite
side
is
constant.
 
 
 
 
 
 
 
 
 
 
 
 sin A sin B sin C 
 = = a b c 
 
 Proof:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Note:

To
use
law
of
sines,
you
will
need
the
measure
of
one
angel
and
its
opposite
 side
plus
another
angel
or
side.
 
 Ex:

Suppose
a=4cm,
b=9cm
and
<B=600.

Find
the
angels
<A,

<C,
and
side
c
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Given
 < C = 450 , b = 4 2 ft , c = 8ft .


Solve
the
triangel.
 
 Note:

Is
it
possible
for
this
triangle
to
have
two
solutions?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Find
the
lengths
a,
b,
c,
and
d
in
the
following
figure
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 The
Law
of
cosines:

In
any
triangle
(acute,
right
or
obtuse)
the
square
of
the
length
 of
any
side
equals
the
sum
of
the
squares
of
the
lengths
of
the
other
minus
twice
the
 product
of
the
lengths
of
the
other
two
sides
times
the
cosine
of
their
included
 angels.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a 2 = b 2 + c 2 − 2bc cos A b 2 = a 2 + c 2 − 2ac cos B 
 c 2 = a 2 + b 2 − 2ab cos C 
 Note:

1)
All
three
equations
follow
the
same
pattern
 
 (square
of
the
length
of
one
side
)2=
(sum
of
the
square
of
the
other
two
sides)
–

 
 (twice
the
product
of
the
lengths
of
the
other
two
sides
times
the
cosine
of
their
 included
angel)
 
 
 2)

Pythagorian
Theorem
is
a
special
case
of
the
law
of
cosine.
 
 For
example,
let
<A
=
900

Then
the
triangle
becomes
a
right
triangle

 
 Plugging
into
the
law
of
cosine
yields:
 
 a 2 = b 2 + c 2 − 2bcco2 s( 900 ) = b 2 + c 2 
 a 2 = b 2 + c 2 (The conclusion of the Pythagorian Theorem) 
 
 Proof
for
the
Law
of
cosine:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Note:

Law
of
cosine
can
be
used
when
you
know
ASS
or
SSS
of
a
triangle.
 
 Ex:

<A
=
550
,
b
=
9,
c=7.

Find
all
other
angels
and
sides.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Given
a
=
33,
b
=
7,
and
c
=
37.

Solve
the
triangle
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Remember:

No
ambiguous
case
for
the
law
of
cosine
since
cosx
is
positive
in
 quadrant
I
and
IV
and
quadrant
IV
angels
are
greater
than
2700
which
is
not
 possible
for
a
triangel.
 ...
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This note was uploaded on 02/08/2011 for the course MATH 3 taught by Professor Staff during the Winter '08 term at UCSC.

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