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Convert in the polar form 1+7i/(1-3i)2.
Answers
Answer:
what is polar from anyways there u go
Step-by-step explanation:
Solution
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(i) Here, z=
(2−i)
2
1+7i
=
4+i
2
−4i
1+7i
=
4−1−4i
1+7i
=
3−4i
1+7i
×
3+4i
3+4i
=
3
2
+4
2
3+4i+21i+28i
2
=
3
2
+4
2
3+4i+21i−28
=
25
−25+25i
=−1+i
Let rcosθ=−1 and rsinθ=1
On squaring and adding, we obtain
r
2
(cos
2
θ+sin
2
θ)=1+1
⇒r
2
(cos
2
θ+sin
2
θ)=2
⇒r
2
=2 (∵cos
2
θ+sin
2
θ=1)
⇒r=
2
(As r>0 )
∴
2
cosθ=−1 and
2
sinθ=1
⇒cosθ=
2
−1
and
sinθ=
2
1
∴θ=π−
4
π
=
4
3π
(As θ lies in II quadrant )
∴z=rcosθ+irsinθ
=
2
cos
4
3π
+i
2
sin
4
3π
=
2
(cos
4
3π
+isin
4
3π
)
This is the required polar form.
(ii) Here,
z=
1−2i
1+3i
=
1−2i
1+3i
×
1+2i
1+2i
=
1+4
1+2i+3i−6
=
5
−5+5i
=−1+i
Let rcosθ=−1 and rsinθ=1
On squaring and adding, we obtain
r
2
(cos
2
θ+sin
2
θ)=1+1
⇒r
2
(cos
2
θ+sin
2
θ)=2
⇒r
2
=2 (∵cos
2
θ+sin
2
θ=1)
⇒r=
2
(As r>0 )
∴
2
cosθ=−1 and
2
sinθ=1
⇒cosθ=
2
−1
and
⇒sinθ=
2
1
∴θ=π−
4
π
=
4
3π
(As θ lies in II quadrant )
∴z=rcosθ+irsinθ
=
2
cos
4
3π
+i
2
sin
4
3π
=
2
(cos
4
3π
+isin
4
3π
)
This is the required polar form.