QQuestionMathematics
QuestionMathematics
Convert the following complex number into its polar representation:
3i
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Answer
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Step 1:I'll solve this step-by-step using the specified LaTeX formatting guidelines:
Step 2:: Identify the Complex Number
The complex number is $$3i$$, which is in the form of a pure imaginary number.
Step 3:: Convert to Rectangular Form
- Imaginary part ($$b$$) = 3
- Real part (a) = 0
Step 4:: Calculate the Magnitude (Modulus)
The magnitude is calculated using $$r = \sqrt{a^{2} + b^{2}}
r = \sqrt{0^{2} + 3^{2}} = \sqrt{9} = 3
Step 5:: Calculate the Argument (Angle)
The argument $$\theta$$ is calculated using $$\tan^{-1}\left(\frac{b}{a}\right)
Since a = 0 and b = 3, \theta = \frac{\pi}{2} radians or 90 degrees
Step 6:: Polar Representation
The polar form is $$r(\cos\theta + i\sin\theta)$$ or $$r \angle \theta
Final Answer
The polar representation is 3 \angle \frac{\pi}{2} or 3(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2}))
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