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Byju's Answer
Standard XII
Mathematics
Sin(A+B)Sin(A-B)
If A,B,C be t...
Question
If A,B,C be the angles of triangle ABC,such that A =
π
4
and
tan
B
tan
C
=
λ
, then prove that
λ
<
(
√
2
−
1
)
2
o
r
λ
>
(
√
2
+
1
)
2
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Solution
Since
A
=
π
4
∴
B
+
C
=
3
π
4
tan
B
tan
C
=
λ
⇒
tan
B
tan
(
3
π
4
−
B
)
=
λ
or
tan
B
(
−
1
−
tan
B
)
1
−
tan
B
=
λ
∴
tan
3
π
4
=
−
1
or
tan
2
B
+
(
1
−
λ
)
tan
B
+
λ
=
0
Δ
≥
0
⇒
(
1
−
λ
)
2
−
4
λ
≥
0
or
λ
2
−
6
λ
+
1
≥
0
Its roots are
a
=
3
−
2
√
2
,
b
=
3
+
2
√
2
(
λ
−
a
)
(
λ
−
b
)
≥
0
,
a
<
b
∴
λ
≤
a
o
r
λ
≥
b
where
a
=
(
√
2
−
1
)
2
,
b
=
(
√
2
+
1
)
2
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Similar questions
Q.
Let A,B,C be three angles such that
A
=
π
4
and
tan
B
tan
C
=
p
. Find all possible values of p such that A,B,C are the angles of a triangle.
Q.
Assertion :If
A
,
B
,
C
are the angles of a triangle such that angle
A
is obtuse then
tan
B
tan
C
>
1
. Reason: In any triangle ABC
tan
A
=
tan
B
+
tan
C
tan
B
tan
C
−
1
.
Q.
Assertion :Statement 1: If
A
,
B
,
C
are the angles of a triangle such that angle
A
is obtuse,then
tan
B
tan
C
>
1
Reason: Statement 2: In any triangle,
tan
A
=
tan
B
+
tan
C
tan
B
tan
C
−
1
Q.
If
A
,
B
,
C
are the angles of a triangle such that the angle
A
is obtuse, then
tan
B
tan
C
<
Q.
A
B
C
is a variable triangle such that
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)
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and
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lie on line
y
=
x
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λ
(where
λ
is a variable), then locus of the orthocentre of
△
A
B
C
is
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Sin(A+B)Sin(A-B)
Standard XII Mathematics
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