The Law of Sines
•
Section 2.1
43
If
C
is obtuse, then 180
◦
−
C
is acute, as are
A
and
B
. If
A
>
180
◦
−
C
then
A
+
C
>
180
◦
,
which is impossible. Thus, we must have
A
≤
180
◦
−
C
. Likewise,
B
≤
180
◦
−
C
. So by what we
showed above for acute angles, we know that sin
A
≤
sin (180
◦
−
C
) and sin
B
≤
sin (180
◦
−
C
).
But we know from Section 1.5 that sin
C
=
sin (180
◦
−
C
). Hence, sin
A
≤
sin
C
and sin
B
≤
sin
C
when
C
is obtuse.
Thus, sin
A
≤
sin
C
if
C
is acute or obtuse, so by the Law of Sines we have
a
c
=
sin
A
sin
C
≤
sin
C
sin
C
=
1
⇒
a
c
≤
1
⇒
a
≤
c
.
By a similar argument,
b
≤
c
. Thus,
a
≤
c
and
b
≤
c
, i.e.
c
is the largest side.
QED
Exercises
For Exercises 19, solve the triangle
△
ABC
.
1.
a
=
10,
A
=
35
◦
,
B
=
25
◦
2.
b
=
40,
B
=
75
◦
,
c
=
35
3.
A
=
40
◦
,
B
=
45
◦
,
c
=
15
4.
a
=
5,
A
=
42
◦
,
b
=
7
5.
a
=
40,
A
=
25
◦
,
c
=
30
6.
a
=
5,
A
=
47
◦
,
b
=
9
7.
a
=
12,
A
=
94
◦
,
b
=
15
8.
a
=
15,
A
=
94
◦
,
b
=
12
9.
a
=
22,
A
=
50
◦
,
c
=
27
10.
Draw a circle with a radius of 2 inches and inscribe a triangle inside the circle. Use a ruler and
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 Fall '11
 Dr.Cheun
 Calculus, Angles, Sin, Sines

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