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11th grade Complex numbers in polar form worksheet

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Complex numbers in polar form

Grade 11 Math Worksheet - Challenge Me

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Circle the best answer, or write your answer on the line.

  1. Pia changes z = 1 − √3i to polar form with θ in radians, from 0 up to 2π. Which is the polar form of z?

    • 2(cos 5π/6 + i sin 5π/6)
    • 2(cos π/3 + i sin π/3)
    • 2(cos 5π/3 + i sin 5π/3)
    • 4(cos 5π/3 + i sin 5π/3)

  2. Tala changes z = -√3 + i to polar form with θ in radians, from 0 up to 2π. Which is the polar form of z?

    • 2(cos 5π/3 + i sin 5π/3)
    • 2(cos π/6 + i sin π/6)
    • 2(cos 5π/6 + i sin 5π/6)
    • 2(cos 7π/6 + i sin 7π/6)

  3. Zee changes z = -4 − 4i to polar form with θ in radians, from 0 up to 2π. Which is the polar form of z?

    • 32(cos 5π/4 + i sin 5π/4)
    • 4√2(cos 5π/4 + i sin 5π/4)
    • 4√2(cos π/4 + i sin π/4)
    • 4√2(cos 3π/4 + i sin 3π/4)

  4. Esme changes z = 3√3 + 3i to polar form with θ in radians, from 0 up to 2π. Which is the polar form of z?

    • 6(cos π/3 + i sin π/3)
    • 36(cos π/6 + i sin π/6)
    • 6(cos 11π/6 + i sin 11π/6)
    • 6(cos π/6 + i sin π/6)

  5. Jun changes z = 6(cos 5π/6 + i sin 5π/6) to the form a + bi. Which is z in the form a + bi?

    • 3√3 + 3i
    • -√3/2 + (1/2)i
    • -3√3 + 3i
    • 3 − 3√3i

  6. Luz changes z = 8(cos 7π/4 + i sin 7π/4) to the form a + bi. Which is z in the form a + bi?

    • √2/2 − (√2/2)i
    • -4√2 + 4√2i
    • 4√2 + 4√2i
    • 4√2 − 4√2i

  7. Milo changes z = 10(cos π/3 + i sin π/3) to the form a + bi. Which is z in the form a + bi?

    • 5 + 5√3i
    • 5√3 + 5i
    • 1/2 + (√3/2)i

  8. A drone flies from home to z = 12(cos 150° + i sin 150°) km; east is real and north is imaginary. Noah tracks it. What is b, the north part of z?

  9. A boat sails from the dock to z = 14(cos 210° + i sin 210°) km; east is real and north is imaginary. Pia tracks it. What is b, the north part of z?

  10. A kite drifts from the launch point to z = 16(cos 30° + i sin 30°) m; east is real and north is imaginary. Zee tracks it. What is b, the north part of z?

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11th grade Complex numbers in polar form worksheet – printable
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About this worksheet

Complex numbers in polar form. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can find the modulus r and the argument θ of a + bi in any quadrant.
  • I can write a + bi as r(cos θ + i sin θ), and back, with exact values.
  • I can read a position given in polar form as east and north parts.

Standard: N-CN.B.4 (+)

Words to know

argument θ: The angle from the positive real axis to z, turning counterclockwise.

polar form: z written as r(cos θ + i sin θ).

reference angle: The angle between z and the nearest part of the real axis.

radian: An angle unit: a full turn is 2π radians.

Common mistakes

Added the real and imaginary parts to get the distance from 0. The distance is the long side of a right triangle, so use a² + b² and a square root.

Stopped at a² + b² and gave r² instead of r.

Turned clockwise to find the angle. Angles on the complex plane turn counterclockwise from the positive real axis.

What to say

Ask: did you take the square root at the end?

Questions parents ask

What is the most common mistake with complex numbers in polar form?

Added the real and imaginary parts to get the distance from 0. The distance is the long side of a right triangle, so use a² + b² and a square root.

Are these 11th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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