Math, asked by paraspdeshpande, 8 months ago

46. The argument of (1+i(3)^1/2)÷(1-i(3)^1/2)
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Answered by helping27
0

Answer:

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Answered by Isha20076
1

Answer:

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Step-by-step explanation:

The given complex number is

z=\dfrac{1+\sqrt{3}i}{\sqrt{3}+i}z=

3

+i

1+

3

i

Personalize denominator.

z=\dfrac{1+\sqrt{3}i}{\sqrt{3}+i}\times \dfrac{\sqrt{3}-i}{\sqrt{3}-i}z=

3

+i

1+

3

i

×

3

−i

3

−i

z=\dfrac{\sqrt{3}-i+3i-\sqrt{3}i^2+\sqrt{3}i}{3-i^2}z=

3−i

2

3

−i+3i−

3

i

2

+

3

i

z=\dfrac{\sqrt{3}-i+3i-(-1)\sqrt{3}+\sqrt{3}i}{3-(-1)}z=

3−(−1)

3

−i+3i−(−1)

3

+

3

i

[\because i^2=-1][∵i

2

=−1]

z=\dfrac{2\sqrt{3}+2i}{4}z=

4

2

3

+2i

z=\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}iz=

2

3

+

2

1

i

If a complex number is z=x+iy, x>0,y>0 then

Arg(z)=\tan^{-1}(\dfrac{y}{x})Arg(z)=tan

−1

(

x

y

)

Arg(z)=\tan^{-1} (\dfrac{\dfrac{1}{2}}{\dfrac{\sqrt{3}}{2}})Arg(z)=tan

−1

(

2

3

2

1

)

Arg(z)=\tan^{-1} (\dfrac{1}{\sqrt{3}})Arg(z)=tan

−1

(

3

1

)

Arg(z)=\dfrac{\pi}{3}Arg(z)=

3

π

Therefore, the argument of given complex number is π/3.

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