Polar Form (Complex Numbers — Foundations)

Grade 11th

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| Attempts: 25 | Questions: 20 | Updated: May 20, 2026

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1) Find r for z = 5 − 12i.

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

Explanation

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

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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)

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2) If z = r(cosθ + i sinθ), what are the rectangular coordinates?

X = r sinθ and y = r cosθ

X = r cosθ and y = r sinθ

X = cosθ and y = sinθ

X = r tanθ and y = r

The real part x equals r cosθ and the imaginary part y equals r sinθ. Option A swaps sine and cosine. Option C omits the scaling by r. Option D uses tangent which is not part of the conversion formula.

Explanation

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3) If r=5 and θ=60°, what is the rectangular form?

5 + 5i

3 + 4i

2.5 + 4.33i

4 + 3i

x = r cosθ = 5 times 0.5 = 2.5. y = r sinθ = 5 times (√3/2) ≈ 4.33. Rectangular form = 2.5 + 4.33i. Option A uses wrong trig values. Option B gives a 3-4-5 triangle value. Option D reverses x and y.

Explanation

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4) What is the argument of z = 5i?

0°

45°

90°

180°

5i lies on the positive imaginary axis at point (0,5). The angle from the positive real axis to the positive imaginary axis is exactly 90°. Option A gives 0° which is the positive real axis. Option B gives 45° which bisects quadrant I. Option D gives 180° which is the negative real axis.

Explanation

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5) The modulus of z = −7 is _____.

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

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

R = 6, θ = 330°

R = 3, θ = 150°

R = 6, θ = 240°

R = 3, θ = 300°

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°.

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

R = √2, θ = 45°

R = √2, θ = 315° (or −45°)

R = 2, θ = 60°

R = 1, θ = 270°

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

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

45°

315° (or −45°)

135°

225°

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

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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9) Convert 3 + 3i to polar form.

R = 6, θ = 30°

R = 3, θ = 60°

R = 3√2, θ = 45°

R = √3, θ = 60°

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°).

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

2 + 2i

2 + 3.46i

4 + 2i

3 + 2i

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.

Explanation

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11) Write z = −2i in polar form.

R = 2, θ = 90°

R = 2, θ = 0°

R = 2, θ = 180°

R = 2, θ = 270°

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

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

III

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

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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13) What is the polar form of z = -4?

R=4, θ=0°

R=4, θ=90°

R=4, θ=270°

R=4, θ=180°

z = -4 lies on the negative real axis. r =

Explanation

z = -4 lies on the negative real axis. r =

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14) Express z = 3(cos180° + i sin180°) in rectangular form.

-3i

-3

cos180° = -1 and sin180° = 0. So z = 3(-1 + 0i) = -3. The point lies on the negative real axis with no imaginary component. Option A gives +3 with the wrong sign. Options B and C give imaginary results which are incorrect since sin180° = 0.

Explanation

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15) In polar form, z = r cis θ represents what?

R(cosθ - i sinθ)

R(cosθ + i sinθ)

R(cos2θ + i sin2θ)

R(cosθ + sinθ)

The shorthand cis θ stands for cosθ + i sinθ, so r cis θ = r(cosθ + i sinθ). Option A incorrectly uses minus before i. Option C doubles the angle with no justification. Option D omits the imaginary unit i from the sine term entirely.

Explanation

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16) The modulus r of z = x + iy is defined as what?

X + y

X² + y²

√(x - y)

√(x² + y²)

The modulus is the distance from the origin to the point (x,y) in the Argand plane, given by the distance formula √(x²+y²). Option A simply adds the components without squaring or taking a root. Option B gives r² not r. Option C subtracts rather than adding under the square root.

Explanation

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17) The argument θ of z = x + iy is found using what formula with quadrant adjustment?

θ = sin⁻¹(y/x)

θ = cos⁻¹(x/y)

θ = tan⁻¹(y/x) adjusted for quadrant

θ = tan⁻¹(x/y) adjusted for quadrant

The argument uses the ratio of imaginary to real parts: θ = tan⁻¹(y/x). Quadrant adjustment is essential since tan⁻¹ only returns values between -90° and 90°, which can miss the correct quadrant. Option A uses sine incorrectly. Option B inverts the ratio. Option D also inverts y and x.

Explanation

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18) Convert z = 2(cos270° + i sin270°) to rectangular form.

-2

-2i

cos270° = 0 and sin270° = -1. So z = 2(0 + i(-1)) = -2i. The point lies on the negative imaginary axis. Option A gives +2i with the wrong sign. Options B and C give purely real values, but since cos270° = 0 the real part is zero and the result must be purely imaginary.

Explanation

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19) What is the polar form of z = i?

R=1, θ=0°

R=1, θ=45°

R=1, θ=180°

R=1, θ=90°

i corresponds to the point (0,1), one unit from the origin on the positive imaginary axis. r = 1 and θ = 90°. Option A gives 0° which is the positive real axis. Option B gives 45° which bisects quadrant I. Option C gives 180° which is the negative real axis.

Explanation

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