In a triangle ABC if AB = c, BC= a, CA = b, tanA= 1, tanB = 2, tanC = 3 and c=3 then find the area (in sq units) of the triangle ABC.
Answers
Given: tanA:tanB:tanC=1:2:3
⇒
1
tanA
=
2
tanB
=
3
tanC
=k
⇒∑tanA=∏tanA
⇒6k=6k
3
⇒k(1−k
2
)=0
⇒k
=0 and k
2
=1
⇒k=1
∴tanA=1,tanB=2,tanC=3
⇒
cotA
1
=1,
cotB
1
=2,
cotC
1
=3
or cotA=1,cotB=
2
1
,cotC=
3
1
⇒cot
2
A=1,cot
2
B=
4
1
,cot
2
C=
9
1
Add 1 to both sides,we get
⇒1+cot
2
A=2,1+cot
2
B=1+
4
1
=
4
5
,1+cot
2
C=1+
9
1
=
9
10
We know that csc
2
θ=1+cot
2
θ
⇒csc
2
A=2
csc
2
B=
4
5
csc
2
C=
9
10
⇒sin
2
A=
2
1
,sin
2
B=
5
4
,sin
2
C=
10
9
Using sine rul
Given: tanA:tanB:tanC=1:2:3
⇒
1
tanA
=
2
tanB
=
3
tanC
=k
⇒∑tanA=∏tanA
⇒6k=6k
3
⇒k(1−k
2
)=0
⇒k
=0 and k
2
=1
⇒k=1
∴tanA=1,tanB=2,tanC=3
⇒
cotA
1
=1,
cotB
1
=2,
cotC
1
=3
or cotA=1,cotB=
2
1
,cotC=
3
1
⇒cot
2
A=1,cot
2
B=
4
1
,cot
2
C=
9
1
Add 1 to both sides,we get
⇒1+cot
2
A=2,1+cot
2
B=1+
4
1
=
4
5
,1+cot
2
C=1+
9
1
=
9
10
We know that csc
2
θ=1+cot
2
θ
⇒csc
2
A=2
csc
2
B=
4
5
csc
2
C=
9
10
⇒sin
2
A=
2
1
,sin
2
B=
5
4
,sin
2
C=
10
9
Using sine rule, we have
⇒a
2
:b
2
:c
2
=
2
1
:
5
4
:
10
9
=5:8:9