Express the complex number in the form a +ib. Also, find the modulus and conjugate. 1+7i / (2 +i)²
Answers
Step-by-step explanation:
=
(2−i)
2
1+7i
x
(2+i)
2
(2+i)
2
=
(2−i)
2
(2+i)
2
(1+7i)(2+i)
2
=
((2−i)(2+i))
2
(1+7i)(4−1+4i)
=
((2−i)(2+i))
2
(4−1+4i+28i−7i−28)
=
(2
2
+1)
2
(−25+25i)
[(a+ib)(a−ib)=a
2
+b
2
]
=
25
(−25+25i)
=−1+i
If θ is principal argument and r is magnitude of complex number z then Polar form is represented by:
z=r(cosθ+isinθ)
On comparision:
−1=rcosθ and 1=rsinθ
On squaring and adding we get:
r
2
(cos
2
θ+sin
2
θ)=(−1)
2
+1
2
=2
r
2
=2 [cos
2
θ+sin
2
θ=1]
r=
2
further
rcosθ
rsinθ
=
−1
1
tanθ=−1=−tan(
4
π
)
θ=
4
3π
[tan(π−θ)=−tanθ]
Polar representation of the given complex no. is:
(2−i)
2
1+7
=−1+i=
2
(cos(
4
3π
)+isin(
4
3π ))
Answer:
complex no into a+bi form