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Question

If 1+3=reιθ, then θ is equal


A

π3

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B

-π3

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C

2π3

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D

-2π3

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Solution

The correct option is C

2π3


To find the value of θ:

Step 1: Calculate the value of r:

The given equation is 1+3=reιθ

1+3=reιθ1+-3=r(cosθ+ιsinθ)-1+3ι=r(cosθ+ιsinθ)rcosθ=1rsinθ=3Squaring and addingr2(cos2θ+sin2θ)=1+3r2=4(cos2θ+sin2θ=1)r=2

Step 2: Calculate the value of θ

cosθ=12sinθ=32θliesinsecondquadrantθ=ππ3=2π3

Hence option C: 2π3 is the correct option.


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