Lesson Worksheet: Operations on Complex Numbers in Polar Form Mathematics • 12th Grade

In this worksheet, we will practice performing calculations with complex numbers in polar form.

Q1:

Given that 𝑧=2𝜋6+𝑖𝜋6cossin and 𝑧=13𝜋3+𝑖𝜋3cossin, find 𝑧𝑧.

  • A233𝜋2+𝑖𝜋2cossin
  • B23311𝜋6+𝑖11𝜋6cossin
  • C2+13𝜋2+𝑖𝜋2cossin
  • D2+1311𝜋6+𝑖11𝜋6cossin
  • E233𝜋2+𝑖𝜋2cossin

Q2:

Given that 𝑧=55𝜋6+𝑖5𝜋6cossin and 𝑧=4(180+𝑖180)cossin, determine 𝑧𝑧.

  • A9(330+𝑖330)cossin
  • B20(330+𝑖330)cossin
  • C20330+𝑖330cossin
  • D9(330𝑖330)cossin
  • E20(30+𝑖30)cossin

Q3:

If 𝑍=7(𝜃+𝑖𝜃)cossin, 𝑍=16(𝜃+𝑖𝜃)cossin, and 𝜃+𝜃=𝜋, then what is 𝑍𝑍?

  • A112𝑖
  • B112
  • C112𝑖
  • D112

Q4:

Given that 𝑧=2((5𝑎2𝑏)+𝑖(5𝑎2𝑏))cossin and 𝑧=4((4𝑎3𝑏)+𝑖(4𝑎3𝑏))cossin, find 𝑧𝑧.

  • A6((20𝑎+6𝑏)+𝑖(20𝑎+6𝑏))cossin
  • B12((𝑎+𝑏)+𝑖(𝑎+𝑏))cossin
  • C8((20𝑎+6𝑏)+𝑖(20𝑎+6𝑏))cossin
  • D8((9𝑎5𝑏)+𝑖(9𝑎5𝑏))cossin
  • E6((9𝑎5𝑏)+𝑖(9𝑎5𝑏))cossin

Q5:

Given that 𝑧=20𝜋2+𝑖𝜋2cossin and 𝑧=4𝜋6+𝑖𝜋6cossin, find 𝑧𝑧 in polar form.

  • A52𝜋3+𝑖2𝜋3cossin
  • B16𝜋3+𝑖𝜋3cossin
  • C5𝜋2+𝑖𝜋2cossin
  • D5𝜋3+𝑖𝜋3cossin
  • E80𝜋3+𝑖𝜋3cossin

Q6:

Given that 𝑍=5(5𝜃+𝑖5𝜃)cossin, 𝑍=4𝜃+𝑖4𝜃cossin, tan𝜃=43, and 𝜃0,𝜋2, find 𝑍𝑍.

  • A3+4𝑖
  • B4+3𝑖
  • C35+45𝑖
  • D45+35𝑖

Q7:

Given that 𝑧=7𝜋6+𝑖7𝜋6cossin, find 1𝑧.

  • Acossin𝜋6+𝑖𝜋6
  • Bcossin5𝜋6+𝑖5𝜋6
  • Ccossin7𝜋6+𝑖7𝜋6
  • Dsincos5𝜋6+𝑖5𝜋6

Q8:

Consider the complex number 𝑧=1+3𝑖.

Find the modulus of 𝑧.

Find the argument of 𝑧.

  • A2
  • B𝜋6
  • C𝜋3
  • D10
  • E2𝜋3

Hence, use the properties of multiplication of complex numbers in polar form to find the modulus and argument of 𝑧.

  • Amodulus = 8, argument = 𝜋
  • Bmodulus = 10, argument = 𝜋
  • Cmodulus = 8, argument = 𝜋2
  • Dmodulus = 3, argument= 𝜋
  • Emodulus = 10, argument = 𝜋2

Hence, find the value of 𝑧.

Q9:

Given that |𝑍|=2 where principal argument (𝑍)=6𝑎+5𝑏, and |𝑍|=6 where principal argument (𝑍)=6𝑎+4𝑏, find 𝑍𝑍.

  • A12((36𝑎+20𝑏)+𝑖(36𝑎+20𝑏))cossin
  • B8((12𝑎+9𝑏)+𝑖(12𝑎+9𝑏))coscos
  • C12((12𝑎+9𝑏)+𝑖(12𝑎+9𝑏))cossin
  • D8((12𝑎+9𝑏)+𝑖(12𝑎+9𝑏))cossin
  • E8((36𝑎+20𝑏)+𝑖(36𝑎+20𝑏))cossin

Q10:

Given that 𝑧=6(4𝜃+𝑖4𝜃)cossin and 𝑧=13(2𝜃+𝑖2𝜃)sincos, where 0<𝜃<90, determine the trigonometric form of 𝑧𝑧.

  • A2((902𝜃)+𝑖(902𝜃))cossin
  • B2(2𝜃+𝑖2𝜃)cossin
  • C193(2𝜃+𝑖2𝜃)cossin
  • D2((90+2𝜃)+𝑖(90+2𝜃))cossin
  • E193((90+2𝜃)+𝑖(90+2𝜃))cossin

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