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