Question

hey guys i really need help with precal!! asap thanks;)

Answer 1

anyone?

Answer 2

First convert to trigonometric form. Do you know how to do this?

Answer 3

convert 1 - i*sqrt(3) to trigonometric form

Answer 4

ya i dont know how to do that

Answer 5

do you mean polar form?

Answer 6

In general, if you have the complex number x+i*y, then it converts to the trigonometric form r*(cos(theta) + i*sin(theta))

where

r = sqrt(x^2 + y^2)

theta = arctan(y/x)

Answer 7

yes that's one way to think about it

Answer 8

\[r\left(\cos(\theta)+i\sin(\theta)\right)\]
\[r=\sqrt{a^2+b^2}\] \[\tan(\theta)=\frac{b}{a}\]

in your example \(a=1,b=-\sqrt{3}\)

Answer 9

So in your case, x = 1 and y = -sqrt(3), which means..

r = sqrt(x^2 + y^2)

r = sqrt(1^2 + (-sqrt(3))^2)

r = sqrt(1 + 3)

r = sqrt(4)

r = 2

I'll let you find theta

Answer 10

hhow do you know what a and b are

Answer 11

oh i see!

Answer 12

a+bi matches with 1-i*sqrt(3) or 1+(-sqrt(3))*i

Answer 13

\(1-\sqrt{3}i=a+bi\) match them up

Answer 14

lol we even use the same words!

Answer 15

spooky...

Answer 16

lol

Answer 17

tell us what you got for theta

Answer 18

.52

Answer 19

your answer should be in terms of \(\pi\)

Answer 20

perhaps it might be better to write
\[\cos(\theta)=\frac{1}{2},\sin(\theta)=-\frac{\sqrt{3}}{2}\]

Answer 21

i put in my calculater tan^-1(-sqrt 3/ 1)

Answer 22

where on the unit circle do you see the coordinate \((\frac{1}{2},-\frac{\sqrt{3}}{2})\)?

Answer 23

300

Answer 24

300 degrees

Answer 25

convert that to radians

Answer 26

ok fine , that is degrees. in radians?

Answer 27

Great minds...

Answer 28

5pi/3

Answer 29

so

1 - i*sqrt(3)

becomes

2*( cos(5pi/3) + i*sin(5pi/3) )

Answer 30

got it!
now you have to do two more things to find \((1+\sqrt{3}i)^3\)

Answer 31

cube the 2, and multiply the angle by 3

Answer 32

usual typo, i meant \((1-\sqrt{3}i)^3\)

Answer 33

so 8

Answer 34

yes, and...

Answer 35

8(cos 5pi/3) + i sin 5pi/3

Answer 36

no you also have to multiply the angle by 3

Answer 37

so 5 pi?

Answer 38

yes, so you then get

8*( cos(5pi) + i*sin(5pi) )

Answer 39

yes thats not an answer though:/

Answer 40

keep in mind that angles such as 0 and 2pi are coterminal angles

Answer 41

so cos(0) = cos(2pi)

Answer 42

so what is 5pi coterminal to

Answer 43

pi

Answer 44

so

8*( cos(5pi) + i*sin(5pi) )

is the same as

8*( cos(pi) + i*sin(pi) )

Answer 45

ah thank you so much i have one more!

Answer 46

you're welcome

Answer 47

since you're given a set number of choices, you can go through each choice and cube it (using de moivre's theorem) just like we did above

Answer 48

if the cubed answer choice gives you the original problem, then that answer choice is correct