12.6.3 Practice: Complex Numbers Practice
Answer the following questions using what you've learned from this unit. Write your responses in the space
provided.
For questions 1 - 3, the polar coordinates of a point are
...
12.6.3 Practice: Complex Numbers Practice
Answer the following questions using what you've learned from this unit. Write your responses in the space
provided.
For questions 1 - 3, the polar coordinates of a point are given. Find the rectangular coordinates of each point. (2
points each)
1.
X = rcos ø = 2 cos 3pi/4 = 2 cos pi/2 + pi/4 = -2 sin pi/4 = -2(1/-√2) = --√2
Y = rsinø = 2 sin 3pi/4 = 2 sin pi/2 + pi/4 = 2 cos pi/4 = 2(1/√2) = √2
(-√2, √2)
2.
X = rcosø = -4cos 7pi/6 = -4cos (pi + pi/6) = -4(-cos pi/6) = 4cospi/6 = 4(√3/2) = 2√3
Y = rinsinø = -4 sin 7pi/6 = -4sin (pi + pi/6) = -4(-sin pi/6) = 4sinpi/6 = 4 (½) = 2
( 2√3, 2)
3.
X = rcosø =2/3cos (-2pi/3) = ⅔(-½) = -⅓
Y = rsinø = 2/3sin (-2pi/3) =⅔ (-√3/2) = -√3/3
( -⅓, -√3/3)
For questions 4 - 6, the rectangular coordinates of a point are given. Find the polar coordinates of each point. (2
points each)
4. (4, 0)
5. (3, 4)
6. (2, -2)
7. Give two sets of polar coordinates that could be used to plot the given point. (4 points)
a.
b.
8. Identify, graph, and state the symmetries for each polar equation. Write the scale that you are using for the
polar axis. (4 points)
a. r = 9cos(5 )
b. r = 2cos
9. Transform each polar equation to an equation in rectangular coordinates and identify its shape. (4 points)
a. = 1.34 radians
b. r = tan sec
10. Compute the modulus and argument of each complex number. (4 points)
a. -5i
b.
11. Write each complex number in rectangular form. Plot and label (with a - d) each point on the polar axes
below. (4 points)
a. b.
c. d.
12. Let z = 13 + 7i and w = 3(cos(1.43) + isin(1.43)). (6 points)
a. Convert z to polar form.
if z= a+bi then in polar form z can be written as z = |r| (cos(θ) + isin(θ)). according to demoivre's theorem :-
where and θ = tan-1(b/a). so z = 13+7i =
b. Calculate zw using De Moivre's theorem.
c. Calculate using De Moivre's theorem.
For questions 13 - 15, let and . Calculate the following, keeping your answer in polar form.
(2 points each)
13.
z1= 2 e^îpi/5
z2= 8 e^i7pi/6
14.
z2 = 8( cos 7pi/6-i sin 7 pi/6)
15.
z1/z2= 1/4( cos( 29/30)pi-isin(29/30) pi)
For questions 16 - 19, write each expression in the standard form for the complex number a + bi. ( 2 points each)
16.
32 + i (32 sqrt 3)
17.
1/32
18. The complex fifth roots of .
10^(1/5)[cos(11pi/30 + 8pi/5) + i sin(11pi/30 + 8pi/5)]
19. Find all seventh roots of unity and sketch them on the axes below.
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