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Precalculus practice

Complex Numbers in Polar Form

Convert polar and rectangular forms, multiply and divide, apply De Moivre's theorem, and list all roots through fourth degree.

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Use exact modulus and standard angles. For roots, list every distinct root around one complete turn.

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Learning objective

What this activity builds

Use modulus and argument to perform exact polar-form operations and display every distinct nth root.

Worked example

Find the cube roots of 1.

Arguments 0, 2pi/3, and 4pi/3, all with modulus 1.

Divide theta+2kpi by 3 for k=0,1,2.

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  • Multiply moduli and add arguments for products.
  • Reduce arguments only after the operation.
  • Count roots before writing them to avoid omissions.

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Math SheetNameScoreTeacherDateComplex Numbers in Polar FormUse exact modulus and standard angles. For roots, list every distinct root around one complete turn.1. Divide 12[cos(0) + i sin(0)] and 4[cos(0) + i sin(0)].Assumptions: Arguments are coterminal modulo one full turn. Trigonometric-form arguments use the selected angle unit. Divisors are nonzero, powers are at most 6, and every requested root islisted.2. Convert sqrt(2)e^(i pi/4) to a+bi form.Assumptions: Arguments are coterminal modulo one full turn. The exponent uses the radian argument; when degrees are selected, the equivalent degree argument is shown in brackets. Divisorsare nonzero, powers are at most 6, and every requested root is listed.Show exact work and preserve every stated domain restriction.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateComplex Numbers in Polar FormUse exact modulus and standard angles. For roots, list every distinct root around one complete turn.3. Convert z = 0 + 6i to trigonometric polar form.Assumptions: Arguments are coterminal modulo one full turn. Trigonometric-form arguments use the selected angle unit. Divisors are nonzero, powers are at most 6, and every requested root islisted.4. Use De Moivre's theorem to find [9e^(i 5pi/3)]^4.Assumptions: Arguments are coterminal modulo one full turn. The exponent uses the radian argument; when degrees are selected, the equivalent degree argument is shown in brackets. Divisorsare nonzero, powers are at most 6, and every requested root is listed.Show exact work and preserve every stated domain restriction.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateComplex Numbers in Polar FormUse exact modulus and standard angles. For roots, list every distinct root around one complete turn.5. Find all 4th roots of 4096[cos(5pi/6) + i sin(5pi/6)].Assumptions: Arguments are coterminal modulo one full turn. Trigonometric-form arguments use the selected angle unit. Divisors are nonzero, powers are at most 6, and every requested root islisted.6. Multiply 7e^(i pi) and 5e^(i pi).Assumptions: Arguments are coterminal modulo one full turn. The exponent uses the radian argument; when degrees are selected, the equivalent degree argument is shown in brackets. Divisorsare nonzero, powers are at most 6, and every requested root is listed.Show exact work and preserve every stated domain restriction.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateComplex Numbers in Polar FormUse exact modulus and standard angles. For roots, list every distinct root around one complete turn.7. Use De Moivre's theorem to find [11[cos(3pi/2) + i sin(3pi/2)]]^6.Assumptions: Arguments are coterminal modulo one full turn. Trigonometric-form arguments use the selected angle unit. Divisors are nonzero, powers are at most 6, and every requested root islisted.8. Divide 10e^(i 3pi/2) and 4e^(i 3pi/2).Assumptions: Arguments are coterminal modulo one full turn. The exponent uses the radian argument; when degrees are selected, the equivalent degree argument is shown in brackets. Divisorsare nonzero, powers are at most 6, and every requested root is listed.Show exact work and preserve every stated domain restriction.math-sheet.comPage 4 of 4

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