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NDA Math Complex Numbers
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© examsnet.com
Question : 52
Total: 66
What is the real part of
(
sinx
+
i
cosx
)
3
where
i
=
√
−
1
?
[NDA I 2015]
−
c
o
s
3
x
−
s
i
n
3
x
s
i
n
3
x
c
o
s
3
x
Validate
Solution:
Applying Euler’s equation to the above question, we get
z = sin x + i cos x
=
c
o
s
(
π
2
−
x
)
+
i
s
i
n
(
π
2
−
x
)
=
e
i
(
π
2
−
x
)
Now by substituting the above value in the given question, we get
z
3
=
(
s
i
n
x
+
i
c
o
s
x
)
3
=
(
e
i
(
π
2
−
x
)
)
3
=
(
e
i
π
2
e
−
i
x
)
3
=
(
i
e
−
i
x
)
3
=
−
i
e
−
i
3
x
=
−
i
(
c
o
s
3
x
−
i
s
i
n
3
x
)
=
i
(
−
c
o
s
3
x
+
s
i
n
3
x
)
=
−
s
i
n
3
x
−
i
c
o
s
3
x
R
e
(
z
)
=
−
s
i
n
3
x
© examsnet.com
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