Question

the modules and amplitude of the complex number √3+2i is...​

Answer 1

Complex number: z=x+iy
Modulus: z=
x
2
+y
2

​

Amplitude: θ=tan
−1

∣
∣
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∣
​

a
b
​

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(i)
z=7−5i
⇒∣z∣=
7
2
+(−5)
2

​
=8.6
⇒θ=tan
−1

∣
∣
∣
∣
∣
​

7
−5
​

∣
∣
∣
∣
∣
​
=35.53

(ii)
z=
3
​
+
2
​
i
⇒∣z∣=
3
​

2
+
2
​

2

​
=2.23
⇒θ=tan
−1

∣
∣
∣
∣
∣
∣
​

3
​

2
​

​

∣
∣
∣
∣
∣
∣
​
=39.23

(iii)
z=−8+15i
⇒∣z∣=
(−8)
2
+15
2

​
=17
⇒θ=tan
−1

∣
∣
∣
∣
∣
​

−8
(15)
​

∣
∣
∣
∣
∣
​
=61.92

(iv)
z=−3+3i
⇒∣z∣=
(−3)
2
+3
2

​
=4.24
⇒θ=tan
−1

∣
∣
∣
∣
∣
​

3
(−3)
​

∣
∣
∣
∣
∣
​
=45

(v)
z=−4−4i
⇒∣z∣=
(−4)
2
+(−4)
2

​
=5.65
⇒θ=tan
−1

∣
∣
∣
∣
∣
​

(−4)
(−4)
​

∣
∣
∣
∣
∣
​
=45

(vi)
z=
3
​
−i
⇒∣z∣=
(
3
​
)
2
+(−1)
2

​
=2
⇒θ=tan
−1

∣
∣
∣
∣
∣
​

(
3
​
)
(−1)
​

∣
∣
∣
∣
∣
​
=30

(vii)
z=3+i(0)
⇒∣z∣=
(3)
2
+(0)
2

​
=3
⇒θ=tan
−1

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​

(3)
(0)
​

∣
∣
∣
∣
∣
​
=0

(viii)
z=1+i
⇒∣z∣=
(1)
2
+(1)
2

​
=1.414
⇒θ=tan
−1

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∣
∣
∣
∣
​

(1)
(1)
​

∣
∣
∣
∣
∣
​
=45

(ix)
z=1+
3
​
i
⇒∣z∣=
(1)
2
+(
3
​
)
2

​
=2
⇒θ=tan
−1

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∣
∣
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∣
∣
​

(1)
(
3
​
)
​

∣
∣
∣
∣
∣
∣
​
=60

(x)
(1+2i)
2
(1−i)=(−3+4i)(1−i)=−6+7i
z=−6+7i
⇒∣z∣=
(−6)
2
+(7)
2

​
=2
⇒θ=tan
−1

∣
∣
∣
∣
∣
∣
​

(−6)
(
7
​
)
​

∣
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​
=
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