If 90 degrees < x < 180 degrees and sin x = 3/5 then the value of sin x/2 =
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Answer:
There is a property of the
tan
function that states:
if
tan
(
x
2
)
=
t
then
sin
(
x
)
=
2
t
1
+
t
2
From here you write the equation
2
t
1
+
t
2
=
3
5
→
5
⋅
2
t
=
3
(
1
+
t
2
)
→
10
t
=
3
t
2
+
3
→
3
t
2
−
10
t
+
3
=
0
Now you find the roots of this equation:
Δ
=
(
−
10
)
2
−
4
⋅
3
⋅
3
=
100
−
36
=
64
t
−
=
10
−
√
64
6
=
10
−
8
6
=
2
6
=
1
3
t
+
=
10
+
√
64
6
=
10
+
8
6
=
18
6
=
3
Finaly you have to find which of the above answers is the right one. Here is how you do it:
Knowing that
90
°
<
x
<
180
°
then
45
°
<
x
2
<
90
°
Knowing that on this domain,
cos
(
x
)
is a decreasing function and
sin
(
x
)
is an increasing function, and that
sin
(
45
°
)
=
cos
(
45
°
)
then
sin
(
x
2
)
>
cos
(
x
2
)
Knowing that
tan
(
x
)
=
sin
(
x
)
cos
(
x
)
then in our case
tan
(
x
2
)
>
1
Therefore, the correct answer is
tan
(
x
2
)
=
3
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