if tan x = 3/4,π<x<3x/2, find the value of sin x/2 ,cos x/2and tan x /2
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Step-by-step explanation:
Given
π
<
x
<
3
π
2
and
tan
x
=
3
4
π
<
x
<
3
π
2
⇒
π
2
<
x
2
<
3
π
4
→
x
2
∈
2nd quadrant
This means
sin
(
x
2
)
→
+
v
e
cos
(
x
2
)
→
−
v
e
tan
(
x
2
)
→
−
v
e
Now
tan
x
=
3
4
⇒
2
tan
(
x
2
)
1
−
tan
2
(
x
2
)
=
3
4
⇒
8
tan
(
x
2
)
=
3
−
3
tan
2
(
x
2
)
⇒
3
tan
2
(
x
2
)
+
8
tan
(
x
2
)
−
3
=
0
⇒
3
tan
2
(
x
2
)
+
9
tan
(
x
2
)
−
tan
(
x
2
)
−
3
=
0
⇒
3
tan
(
x
2
)
(
tan
(
x
2
)
+
3
)
−
1
(
tan
(
x
2
)
+
3
)
=
0
⇒
(
3
tan
(
x
2
)
−
1
)
(
tan
(
x
2
)
+
3
)
=
0
This means
tan
(
x
2
)
=
1
3
→
not acceptable as
tan
(
x
2
)
→
−
v
e
So
tan
(
x
2
)
→
−
3
Now
cos
(
x
2
)
=
1
sec
(
x
2
)
=
−
1
√
1
+
tan
2
(
x
2
)
=
−
1
√
1
+
(
−
3
)
2
=
−
1
√
10
Again
sin
(
x
2
)
=
tan
(
x
2
)
×
cos
(
x
2
)
=
−
3
×
(
−
1
√
10
)
=
3
√
10
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