Rectangular and Polar Forms (Grades 11-12)

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Rectangular and Polar Forms

1. 
Convert [math]-4+2i[/math] to polar form.
  1. [math] 2sqrt(5) \ [ \ cos(153.4°) + i sin(153.4°) \ ] [/math]
  2. [math] 2sqrt(3) \ [ \ cos(153.4°) + i sin(153.4°) \ ] [/math]
  3. [math] 2 sqrt(5) \ [ \ cos(-26.6°) + i sin(-26.6°) \ ][/math]
  4. [math] 2sqrt(3) \ [ \ cos(-26.6°) + i sin(-26.6°) \ ] [/math]
2. 
What is the polar form of the complex number [math]-5 ?[/math]
  1. [math] -5 \ (cos180° + i sin180°)[/math]
  2. [math]5 \ (cos0° + i sin0°)[/math]
  3. [math]5 \ (cos180° + isin180°)[/math]
  4. [math]"This is not a complex number."[/math]
3. 
What are the polar coordinates of [math]2-5i[/math] to one decimal place?
  1. [math](5, -68deg)[/math]
  2. [math](5.4, 68.2deg)[/math]
  3. [math](5.4, -68.2deg)[/math]
  4. [math](5, 68deg)[/math]
4. 
What are the polar coordinates of [math]-3+7i[/math] to one decimal place?
  1. [math](3.7, -10deg)[/math]
  2. [math](-7.6, -113.1deg)[/math]
  3. [math](76, 113deg)[/math]
  4. [math](7.6, 113.2deg)[/math]
5. 
Convert [math]3sqrt(7) \ (cos80° + i sin80°)[/math] to rectangular form.
  1. [math]7.8 + 1.4i[/math]
  2. [math]1.4 + 1.0 i[/math]
  3. [math]1.4 + 7.8i[/math]
  4. [math]0.2 + 1.0i[/math]
6. 
What is the rectangular form of the complex number whose polar coordinates are [math](6, -18°) ?[/math]
  1. [math]5.7 - 0.3 i[/math]
  2. [math]5.7 - 1.9i[/math]
  3. [math]5.7 + 1.9i[/math]
  4. [math]1.9 - 5.7i[/math]
7. 
Let the following graph represent the complex plane (assume that x is the real axis and y is the imaginary axis). Let [math]z = 4+3i[/math] be the complex number represented by the letter I.
Coordinate Plane - 5x5 - With Dots
A. 
What is the distance from I to the origin? Let [math]alpha[/math] represent this value.
  1. [math]alpha = 5[/math]
  2. [math]alpha = 2sqrt(5)[/math]
  3. [math]alpha = 4[/math]
  4. [math]alpha = 3[/math]
B. 
If a line is drawn between I and the origin, what is the angle between this line and the positive x-axis, starting from the x-axis and going in a counterclockwise direction? Let [math]beta[/math] represent this angle.
  1. [math]beta = 45.0°[/math]
  2. [math]beta = 53.1°[/math]
  3. [math]beta = 36.9°[/math]
  4. [math]beta = 48.6°[/math]
C. 
What are the polar coordinates of the complex number [math]z ?[/math]
  1. [math] (4, 30.2°)[/math]
  2. [math] (5, 36.9°)[/math]
  3. [math] (5, 53.1°)[/math]
  4. [math] (4, 53.1°)[/math]
D. 
What is the significance of the answers in parts a and b compared to part c? Choose the best answer.
  1. They are similar, since the geometry of the complex plane is closely related to the polar coordinates of a complex number.
  2. They are the same. This is only a phenomenon of the first quadrant though; for a complex number in any other quadrant, these values would be different.
  3. They are the same, since [math]alpha[/math] represents the modulus of [math]z[/math], and [math]beta[/math] represents the argument of [math]z.[/math]
  4. Any similarity or equality is pure chance; there is no relationship between these values.

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