prove that a/sinA= b/sinB= c/sinC ?
it is also known as sine formula
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Step-by-step explanation:
Let ,
−−→
A
B
=
¯
c
,
−−→
B
C
=
¯
a
,
−−→
C
A
=
¯
b
So,
¯
a
+
¯
b
+
¯
c
=
¯
0
Using definition of cross Product
¯
a
×
(
¯
a
+
¯
b
+
¯
c
)
=
¯
a
×
¯
0
⇒
(
¯
a
×
¯
a
)
+
(
¯
a
×
¯
b
)
+
(
¯
a
×
¯
c
)
=
¯
0
→
[
∵
¯
a
×
¯
0
=
¯
0
]
⇒
¯
0
+
(
¯
a
×
¯
b
)
+
(
¯
a
×
¯
c
)
=
¯
0
→
[
∵
(
¯
a
×
¯
a
)
=
¯
0
]
⇒
(
¯
a
×
¯
b
)
−
(
¯
c
×
¯
a
)
=
¯
0
⇒
(
¯
a
×
¯
b
)
=
(
¯
c
×
¯
a
)
⇒
a
⋅
b
sin
(
π
−
C
)
=
c
⋅
a
sin
(
π
−
B
)
⇒
b
sin
C
=
c
sin
B
⇒
sin
C
c
=
sin
B
b
...
.
→
(
1
)
Similarly we can prove that ,
⇒
sin
A
a
=
sin
B
b
...
.
→
(
2
)
Hence ,
sin
A
a
=
sin
B
b
=
sin
C
c
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