1.4 Polar Form

• What are the two components used to define the polar form of a complex number?

  Modulus and argument
• The modulus of a complex number is the distance from the origin
• The argument of a complex number is the angle between the positive real axis and the line connecting the origin to the complex number.
• What is the polar form expression for a complex number $z$ with modulus $r$ and argument $\theta$?


  $z =$$r(\cos \theta + i\sin \theta)$
• The modulus of $z =$$3 +$$4i$ is 5
• What is the argument of $z =$$3 +$$4i$ in radians?


  $\theta \approx 0.927$
• What is the formula to calculate the modulus $r$ of a complex number $z =$$x +$$iy$?


  $r =$ \sqrt{x^{2} + y^{2}}
• Steps to convert a complex number from Cartesian form to polar form

  1️⃣ Calculate the modulus $r$

  2️⃣ Calculate the argument $\theta$

  3️⃣ Express in polar form $z =$$r(\cos \theta + i\sin \theta)$
• The argument $\theta$ must be adjusted based on the quadrant of the complex number to ensure it is in the correct range
• The complex number $z =$$- 3 - 4i$ lies in the third quadrant.
• What is the adjusted argument $\theta$ for $z =$$- 3 - 4i$ in radians?


  $\theta \approx - 2.214$
• What is the formula to calculate the real component $x$ of a complex number in polar form?


  $x =$$r\cos \theta$
• The imaginary component $y$ of a complex number in polar form is calculated using $y =$$r\sin \theta$
• Converting $z =$$5(\cos 0.927 + i\sin 0.927)$ to Cartesian form results in $z =$$3 +$$4i$.
• What is the Cartesian form of a complex number $z$?


  $z =$$x +$$iy$
• To convert a complex number from polar form to Cartesian form, you first calculate the real component using the formula $x =$$r\cos \theta$
• The imaginary component $y$ in Cartesian form is calculated as y = r\cos \theta</latex>.


  False
• What is the term for $r =$ \sqrt{x^{2} + y^{2}} in the polar form of a complex number?


  Modulus
• What is the formula to calculate the argument $\theta$ in the polar form of a complex number?


  $\theta =$$\arctan(\frac{y}{x})$
• When converting from Cartesian form to polar form, the argument $\theta$ must be adjusted based on the quadrant the complex number lies in, such as adding $\pi$ in the second quadrant.
• If a complex number lies in the third quadrant, the argument \theta</latex> is adjusted by adding $\pi$.


  False
• What is the formula for multiplying two complex numbers in polar form?

  $r_{1} r_{2} (\cos(\theta_{1} +$$\theta_{2}) +$$i\sin(\theta_{1} +$$\theta_{2}))$
• What is the formula for dividing two complex numbers in polar form?

  \frac{r_{1}}{r_{2}} (\cos(\theta_{1} - \theta_{2}) + i\sin(\theta_{1} - \theta_{2}))</latex>
• What is the formula for multiplying two complex numbers in polar form?

  $r_{1} r_{2} (\cos(\theta_{1} +$$\theta_{2}) +$$i\sin(\theta_{1} +$$\theta_{2}))$
• The formula for dividing two complex numbers in polar form involves subtracting their arguments
• De Moivre's theorem states that $z^{n} =$$r^{n}(\cos n\theta + i\sin n\theta)$ for any integer $n$ and complex number $z =$$r(\cos \theta + i\sin \theta)$.
• What is the modulus of the complex number $1 +$$i$?


  $\sqrt{2}$
• What is the argument of the complex number $1 +$$i$?


  $\frac{\pi}{4}$
• De Moivre's theorem can be used to raise a complex number in polar form to a positive integer
• Steps to calculate $(1 + i)^{5}$ using De Moivre's theorem


  1️⃣ Convert $1 +$$i$ to polar form

  2️⃣ Calculate modulus $r$

  3️⃣ Calculate argument $\theta$

  4️⃣ Apply De Moivre's theorem

  5️⃣ Convert back to Cartesian form
• De Moivre's theorem applies only to complex numbers in polar form.
• What is the modulus of the complex number $1 +$$i$?


  $\sqrt{2}$
• What does the polar form of a complex number use to represent it in the complex plane?

  Modulus and argument
• What is the formula for calculating the modulus $r$ of a complex number $z =$$x +$$iy$?


  $r =$ \sqrt{x^{2} + y^{2}}
• The argument $\theta$ of a complex number $z =$$x +$$iy$ is calculated using the arctan
• Match the quadrant with the condition for adjusting the argument:

  Quadrant I ↔️ $x > 0, y > 0$

  Quadrant II ↔️ $x < 0, y > 0$

  Quadrant III ↔️ $x < 0, y < 0$

  Quadrant IV ↔️ $x > 0, y < 0$
• The polar form of a complex number $z$ is expressed as $z =$$r(\cos \theta + i\sin \theta)$.
• The argument of the complex number $z =$$- 3 - 4i$ in Quadrant III is approximately -2.214 radians.
• The argument of a complex number in Quadrant III is adjusted by subtracting π.
• What is the polar form of the complex number z = -3 - 4i</latex>?

  $5(\cos( - 2.214) +$$i\sin( - 2.214))$