cos\theta +sin\theta +icos\theta express in polar form
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Express the following complex number in polar form and exponential form: (1−cosθ+isinθ),θϵ(0,π)
(1−cosθ+isinθ),θ∈(0,π)
⇒1−cosθ+isinθ=x(cosθ+isinθ)
⇒tanθ=
1−cosθ
sinθ
=
2sin
2
2
θ
2sin
2
θ
cos
2
θ
=cot
2
θ
⇒θ=tan
−1
(cot
2
θ
)
∴θ=
2
π
−
2
θ
⇒r=
(1−cosθ)
2
+sin
2
θ
=
2−2cosθ
=2sin
2
θ
⇒1−cosθ+isinθ=2sin
2
θ
e
(π/2−θ/2)i
⇒2sin(
2
θ
)[cos(
2
π
−
2
θ
)+isin(
2
π
−
2
θ
)];2sin
2
θ
e
(π/2−θ/2)i
Hence, the answer is 2sin(
2
θ
)[cos(
2
π
−
2
θ
)+isin(
2
π
−
2
θ
)];2sin
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