Polar Form (Complex Numbers — Foundations)

  • 11th Grade
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| Questions: 20 | Updated: Dec 17, 2025
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1) Convert 3 + 3i to polar form.

Explanation

The modulus r = √(3² + 3²) = √18 = 3√2. The argument θ = tan⁻¹(3 / 3) = 45°. Thus, z = 3√2(cos 45° + i sin 45°).

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About This Quiz
Polar Form (Complex Numbers Foundations) - Quiz

Curious about representing complex numbers in a new and more visual way? In this quiz, you’ll learn how polar form captures magnitude and direction, and how to convert seamlessly between rectangular and polar coordinates. You’ll practice interpreting complex numbers as points on the plane, connect angles to arguments, and see... see morewhy polar form becomes especially powerful in multiplication and division. Each question builds your comfort with this essential representation.
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2) Find r for z = 5 − 12i.

Explanation

 Compute r = √(5² + (−12)²) = √(25 + 144) = √169 = 13. This represents the distance of z from the origin.

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3) Determine the argument θ for z = 4 − 4i.

Explanation

tan θ = (−4 / 4) = −1, so θ = −45°. Since the point (4, −4) lies in quadrant IV, the positive angle is 315°.

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4) Which quadrant does z = −3 + 4i lie in?

Explanation

The real part is negative and the imaginary part positive; hence z lies in the second quadrant (angles between 90° and 180°).

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5) Convert z = 1 − i to polar form (r, θ).

Explanation

r = √(1² + (−1)²) = √2. The point (1, −1) is in quadrant IV, so θ = 315°. Thus z = √2 (cos 315° + i sin 315°).

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6) Write z = −2i in polar form.

Explanation

z lies on the negative imaginary axis, so r = |−2| = 2 and θ = 270°. Therefore z = 2 (cos 270° + i sin 270°).

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7) Find the rectangular form of z = 4 (cos 60° + i sin 60°).

Explanation

To convert z = 4 (cos 60° + i sin 60°) into rectangular form, substitute cos 60° = 0.5 and sin 60° = √3/2 ≈ 0.866 to get z = 4(0.5 + 0.866i) = 2 + 3.464…i ≈ 2 + 3.46i, so the correct choice is B.

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8) Determine r and θ for z = −3 − 3√3 i.

Explanation

r = √(9 + 27) = √36 = 6. tan θ = (−3√3)/(−3) = √3 → θ = 60°, but since both parts are negative, z is in quadrant III, so θ = 180° + 60° = 240°.
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9) The polar form r (cos θ + i sin θ) is also known as:

Explanation

Here r is the modulus and θ the argument. Together they define the modulus–argument form of a complex number.

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10) The modulus of z = −7 is _____.

Explanation

The modulus is the distance from the origin, so r = √(−7)² = 7, meaning the blank should be filled with 7.

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11) The argument of z = 5i is _____.

Explanation

Since 5i lies on the positive imaginary axis, its angle from the positive real axis is 90°, so the blank should contain 90°.

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12) If r = 5 and θ = 60°, find the rectangular form.

Explanation

Using x = r cos θ = 5·0.5 = 2.5 and y = r sin θ = 5·0.866 ≈ 4.33, the rectangular form becomes 2.5 + 4.33i, so the blank should be 2.5 + 4.33i.

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13) If z = r (cos θ + i sin θ), then the rectangular coordinates are _____.

Explanation

The real and imaginary parts are given by x = r cos θ and y = r sin θ, so the blank should be x = r cos θ, y = r sin θ.
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14) Express z = 3 (cos 180° + i sin 180°) in rectangular form.

Explanation

With cos 180° = −1 and sin 180° = 0, the value becomes 3(−1 + 0i) = −3, so the blank should be −3.
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15) The polar form of z = −4 is _____.

Explanation

Because the number lies on the negative real axis, its modulus is 4 and its argument is 180°, so the blank should be r = 4, θ = 180°.
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16) In polar form, z = r cis θ represents _____.

Explanation

The shorthand cis θ expands to cos θ + i sin θ, so the blank should be r (cos θ + i sin θ).
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17) The modulus r of a complex number z = x + iy is _____.

Explanation

The modulus is obtained using the distance formula √(x² + y²), so the blank should be √(x² + y²)
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18) The argument θ of a complex number z = x + iy is _____.

Explanation

The argument comes from θ = tan⁻¹(y/x) with appropriate quadrant adjustments, so the blank should be tan⁻¹(y/x) (adjusted for quadrant).
If using tan⁻¹(y / x), remember to adjust:
– Add 180° if x < 0 and y ≥ 0 (Quadrant II)
– Subtract 180° if x < 0 and y < 0 (Quadrant III)
– Use 90° if x = 0 and y > 0
– Use 270° (or −90°) if x = 0 and y < 0
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19) Convert z = 2 (cos 270° + i sin 270°) to rectangular form.

Explanation

Using cos 270° = 0 and sin 270° = −1, the expression becomes 2(0 + i(−1)) = −2i, so the correct rectangular form is −2i.
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20) The polar form of z = i is _____.

Explanation

Since i corresponds to the point (0, 1) which is one unit above the origin, its modulus is 1 and its argument is 90°, so the blank should be filled with r = 1, θ = 90°.
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Convert 3 + 3i to polar form.
Find r for z = 5 − 12i.
Determine the argument θ for z = 4 − 4i.
Which quadrant does z = −3 + 4i lie in?
Convert z = 1 − i to polar form (r, θ).
Write z = −2i in polar form.
Find the rectangular form of z = 4 (cos 60° + i sin 60°).
Determine r and θ for z = −3 − 3√3 i.
The polar form r (cos θ + i sin θ) is also known as:
The modulus of z = −7 is _____.
The argument of z = 5i is _____.
If r = 5 and θ = 60°, find the rectangular form.
If z = r (cos θ + i sin θ), then the rectangular...
Express z = 3 (cos 180° + i sin 180°) in rectangular form.
The polar form of z = −4 is _____.
In polar form, z = r cis θ represents _____.
The modulus r of a complex number z = x + iy is _____.
The argument θ of a complex number z = x + iy is _____.
Convert z = 2 (cos 270° + i sin 270°) to rectangular form.
The polar form of z = i is _____.
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