Question

Find the modulus and the argument of the following complex numbers : z=1+i✓3​

Answer 1

Answer:

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Step-by-step explanation:

z=−1−i

3

Let rcosθ=−1andrsinθ=−

3

On squaring and adding, we obtain

(rcosθ)

2

+(rsinθ)

2

=(−1)

2

+(−

3

)

2

⇒r

2

(cos

2

θ+sin

2

θ)=1+3

⇒r

2

=4[cos

2

θ+sin

2

θ=1]

⇒r=

4

=2 (Conventionally,r>0 )

∴ Modulus of z i.e ∣z∣=2

∴2cosθ=−1and2sinθ=−

3

Since both the values of sinθ and cosθ are negative and sinθ and cosθ are negative in III quadrant,

Argument =−(π−

3

π

)=

3

−2π

Thus, the modules and argument of the complex number −1−

3i

are 2 and

3

−2π

respectively