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Polar Form of Complex Numbers quiz

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  • What is the polar form of a complex number?
    The polar form is r(cos θ + i sin θ), where r is the distance from the origin and θ is the angle with the real axis.
  • How do you calculate r for a complex number x + yi?
    Use the Pythagorean theorem: r = √(x² + y²), where x and y are the real and imaginary parts.
  • How do you find θ for a complex number x + yi?
    θ is found using tan θ = y/x, then θ = arctan(y/x), adjusting for the correct quadrant.
  • What adjustment must you make to θ in quadrant II or III?
    Add 180° to θ if the complex number is in quadrant II or III.
  • What adjustment must you make to θ in quadrant IV?
    Add 360° to θ if the complex number is in quadrant IV.
  • How do you convert from polar form to rectangular form?
    Distribute r: z = r(cos θ + i sin θ), then evaluate the cosine and sine to find the real and imaginary parts.
  • What is the rectangular form of a complex number?
    The rectangular form is x + yi, where x is the real part and y is the imaginary part.
  • What is the real part of z = 5(cos 37° + i sin 37°)?
    The real part is 5 × cos(37°), which is approximately 4.
  • What is the imaginary part of z = 5(cos 37° + i sin 37°)?
    The imaginary part is 5 × sin(37°), which is approximately 3.
  • How do you convert z = 8(cos π/6 - i sin π/6) to rectangular form using unit circle values?
    Use cos(π/6) = √3/2 and sin(π/6) = 1/2, so z = 4√3 - 4i.
  • What is the value of cos(π/6) from the unit circle?
    cos(π/6) = √3/2.
  • What is the value of sin(π/6) from the unit circle?
    sin(π/6) = 1/2.
  • Why is r always straightforward to calculate in polar form?
    Because r is always found using the Pythagorean theorem, regardless of the quadrant.
  • What is the main strategy for converting from polar to rectangular form?
    Distribute the r value to both cosine and sine, then evaluate the expressions.
  • What is the polar form of 4 + 3i?
    It is 5(cos 37° + i sin 37°), where r = 5 and θ ≈ 37°.