Converting Between Rectangular and Polar Forms of Complex Numbers

  • 12th Grade
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| Attempts: 13 | Questions: 10 | Updated: Dec 10, 2025
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Question 1 / 11
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1) Convert 1 + i to polar form.

Explanation

r = √(1²+1²)=√2 and θ = tan⁻¹(1/1)=45°.

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About This Quiz
Converting Between Rectangular And Polar Forms Of Complex Numbers - Quiz

Complex numbers have two powerful forms—and knowing how to switch between them is key! In this quiz, you’ll practice converting rectangular form to polar form and back again. Take this quiz to master flexibility in representing complex numbers.

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2) Convert 2 + 2√3 i to polar form.

Explanation

r = √(2²+(2√3)²)=√(4+12)=4, θ = tan⁻¹(2√3/2)=tan⁻¹(√3)=60°.

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3) Convert −3 + 3i to polar form.

Explanation

r = √((-3)²+3²)=√(18)=3√2; angle is in quadrant II with tan⁻¹(|3/−3|)=45°, so θ=135°.

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4) Convert polar form r = 5, θ = 0° to rectangular form.

Explanation

x=5cos0°=5, y=5sin0°=0.

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5) Convert polar form r = 10, θ = 180° to rectangular form.

Explanation

x=10cos180°=−10, y=10sin180°=0.

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6) Convert polar form r = 4, θ = 90° to rectangular form.

Explanation

x=4cos90°=0, y=4sin90°=4.

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7) Convert 0 − 5i to polar form.

Explanation

r=5, point is on negative imaginary axis → 270°.

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8) Convert −4 − 4i to polar form.

Explanation

r=√(16+16)=4√2; both x and y negative → quadrant III; tan⁻¹(4/4)=45°, so θ=225°.

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9) Convert polar form r = 6, θ = 45° to rectangular form.

Explanation

x=6cos45°=6(√2/2)=3√2; y=6sin45°=3√2.

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10) Convert polar form r = 8, θ = 300° to rectangular form.

Explanation

x=8cos300°=8(½)=4√3, y=8sin300°=8(−½)=−4.

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Convert 1 + i to polar form.
Convert 2 + 2√3 i to polar form.
Convert −3 + 3i to polar form.
Convert polar form r = 5, θ = 0° to rectangular form.
Convert polar form r = 10, θ = 180° to rectangular form.
Convert polar form r = 4, θ = 90° to rectangular form.
Convert 0 − 5i to polar form.
Convert −4 − 4i to polar form.
Convert polar form r = 6, θ = 45° to rectangular form.
Convert polar form r = 8, θ = 300° to rectangular form.
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