6.5 Complex Numbers in Polar Form: DeMoivre's Theorem

​6.5 Complex Numbers in Polar Form: DeMoivre's Theorem

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Multiple Choice

Find the modulus of the complex number. Also, state what quadrant the point would land in.

-3 + i #28

• √10, Quadrant 2

• √10, Quadrant 3

• 2, Quadrant 2

• 3, Quadrant 3

Multiple Choice

Where is point C located? #10

• 2 + i

• -3

• 2i

• -1 - i

Multiple Choice

What complex number is represented by the graph? #8

• -3-i

• -3+i

• 1+3i

• 1-3i

Multiple Choice

$$\left|a+bi\right|$$  is defined as: #12

• $$\sqrt{a^2+b^2}$$  

• $$\sqrt{a^2-b^2}$$  

• $$\sqrt{a^2+\left(bi\right)^2}$$  

• $$\sqrt{a^2-\left(bi\right)^2}$$  

Multiple Choice

Find the absolute value of the complex number 5 - 2i. #11

• $$\sqrt{21}$$

• $$\sqrt{29}$$

• $$\pm\sqrt{29}$$

• $$\pm\sqrt{3}$$

Multiple Select

Choose all that apply.  Find $$\left|6+8i\right|$$ #13

• $$\sqrt{\left(6\right)^2-\left(8i\right)^2}$$  

• $$\sqrt{\left(6\right)^2+\left(8\right)^2}$$  

• 10

• 10i

Multiple Choice

Write in Polar Form :

$$-1,\ -i\sqrt{3}$$  

#1

• Z=3(COS 30 +I SIN 30)

• Z=2(COS 60 + I SIN 60)

• Z= 4(COS 30 + I SIN 30)

• Z+ 2(COS 30 + I SIN 30)

Multiple Choice

$$WRITE\ IN\ TRIGONOMETRIC\ FORM:\ \ \$$   $$-1\ +i$$  #3

• $$\left(\sqrt{2}\left(\cos\left(\frac{1}{4}\pi\right)\right)\ +\ i\ \sin\ \left(\frac{1}{4}\pi\right)\right)$$  

• $$\sqrt{2}\left(\cos\ \frac{3}{4}\pi\ +i\ \sin\ \frac{3}{4}\pi\right)$$  

• $$\sqrt{3}\left(\sin\ \frac{1}{4}\pi+\ \cos\ \frac{1}{4}\pi\right)$$  

• $$\sqrt{2}\left(\cos\ \frac{5}{4}\pi\ +\ \sin\ \frac{5}{4}\pi\right)$$  

Multiple Choice

$$Write\ in\ Polar\ Form\ :$$   $$3\ +6i$$  #6

• $$3\sqrt{5\ }\left(Cos\ 60\degree\ +\ i\ \sin\ 60\degree\right)$$  

• $$3\sqrt{5}\left(Cos\ 64.4\degree\ \ +\ i\ \sin\ 64.4\degree\right)$$  

• $$3\sqrt{5}\left(Cos\ 63.4\degree\ +\ i\ Sin\ 63.4\degree\right)$$  

• $$3\sqrt{5}\left(Cos\ 65.4\degree\ +\ i\ Sin\ 65.4\degree\right)$$  

Multiple Choice

$$Write\ in\ rec\tan gular\ Form:$$   $$2\left(Cos\ 30\degree+\ i\ Sin\ \ 30\degree\right)$$  #7

• $$3\ +\ 2\ i$$  

• $$\sqrt{3}\ -\ i$$  

• $$\sqrt{3}\ +i$$  

• $$-\sqrt{3}\ +i$$  

Multiple Choice

Write the complex number in rectangular form. 4(cos7π/4 + isin7π/4) #20

• 2√2 - 2√2i

• 2√2 + 2√2i

• √2 + i

• 2i- √2

Multiple Choice

Write the complex number in rectangular form. 2(cos5π/6 + isin5π/6) #21

• -√3 + i

• -√3 - i

• -1 + √3i

• -3i

Multiple Choice

Find zw. Leave your answer in polar form. z=2(cos40 + isin40) w=4(cos20 + isin20) #22

• 8(cos60 + isin60)

• -8(cos20 - isin20)

• 6(sin60 + icos60)

• -4(cos20 +isin20)

Multiple Choice

Find zw. Leave your answer in polar form. z=cos120 +isin120 w=cos100 +isin100 #23

• cos220 + isin220

• cos20 + isin20

• sin220 + icos220

• -cos220 - isin20

Multiple Choice

Find the product of $$z_1=3\left(\cos\ \frac{2\pi}{3}+i\sin\ \frac{2\pi}{3}\right)$$  and

$$z_2=5\left(\cos\ \frac{\pi}{4}+i\sin\ \frac{\pi}{4}\right)$$  . #31

• $$15\left(\cos\ \frac{3\pi}{7}+i\sin\ \frac{3\pi}{7}\right)$$  

• $$15\left(\cos\ \frac{\pi}{6}+i\sin\ \frac{\pi}{6}\right)$$  

• $$15\left(\cos\ \frac{5\pi}{12}+i\sin\ \frac{5\pi}{12}\right)$$  

• $$15\left(\cos\ \frac{11\pi}{12}+i\sin\ \frac{11\pi}{12}\right)$$  

Multiple Choice

Find z/w. Leave your answer in polar form. z=2(cos40 + isin40) w=4(cos20 + isin20) #24

• 1/2 (cos20 + isin20)

• 1/4 (cos20 + isin20)

• 8(cos60 - isin60)

• 2(sin20 + isin60)

Multiple Choice

Find z/w. Leave your answer in polar form. z=cos120 +isin120 w=cos100 +isin100 #25

• cos20 +isin20

• cos220 + isin220

• cos20 -isin20

• 2(cos220 +isin220)

Multiple Choice

Use De Moivre's Theorem to find the indicated power of the complex number $$\left[3\left(\cos\ \frac{\pi}{4}+i\sin\ \frac{\pi}{4}\right)\right]^{10}$$   for $$0\le\theta\le2\pi$$  . #32

• $$3^{10}\left(\cos\ \frac{5\pi}{2}+i\sin\ \frac{5\pi}{2}\right)$$  

• $$3^{10}\left(\cos\ \frac{\pi}{2}+i\ \sin\ \frac{\pi}{2}\right)$$  

• $$3^{10\ }\left(\cos\ \frac{\pi}{4}+\ i\ \sin\ \frac{\pi}{4}\ \right)$$  

• $$3\ \left(\cos\ \frac{\pi}{2}+\ \sin\ \frac{\pi}{2}\right)$$  

Multiple Choice

Find  (3 + 3√3 i )⁴  using De Moivre's Theorem and express your answer in rectangular form. #40

• -648 - 648 √3 i

• -654 - 654 √3 i

• -1962 - 1962 √3 i

• 648 i

Multiple Choice

Which formula should be used if we are given theta in radians? #37

• $$r^{\left(\frac{1}{n}\right)}\left[\cos\left(\frac{\theta+360k}{n}\right)+i\sin\left(\frac{\theta+360k}{n}\right)\right]$$  

• $$r^{\left(\frac{1}{n}\right)}\left[\cos\left(\frac{\theta+2\pi k}{n}\right)+i\sin\left(\frac{\theta+2\pi k}{n}\right)\right]$$  

Multiple Choice

If you are asked to find the cubed roots of  $$64\left(\cos435\degree+i\sin435\degree\right)$$ , how many answers should you have? #38

• 1

• 2

• 3

• 4

Multiple Choice

Which of the following is the correct cubed roots of  #38? #39

• $$4\left(\cos145\degree+i\sin145\degree\right)$$  

  $$4\left(\cos290\degree+i\sin290\degree\right)$$ $$4\left(\cos435\degree+i\sin435\degree\right)$$   

• $$4\left(\cos145\degree+i\sin145\degree\right)$$  

  $$4\left(\cos263\degree+i\sin263\degree\right)$$ $$4\left(\cos383\degree+i\sin383\degree\right)$$