Teach me how to express 5sqrt3 - 5i in polar form please.

I don't want you to do the work for me. Just show me the steps I need to do the work properly on my own. Otherwise I will not pass this class or the exam when I enter college, and I do not want to retake PreCalc.

Its a multiple choice homework question with answer choices:
a: 10(cos 11pi/6 + i sin 11pi/6)
b: 10(cos 11pi/6 - i sin 11pi/6)
c: 5 (cos 11pi/6 + i sin 11pi/6)
d: 10(cos 5pi/3 + i sin 5pi/3)

I did as much work as I could guess at, and came up with r=10sqrt2,
arctan=1/sqrt3 which equals sqrt3/3.I know that sqrt3/3 = tan 30 and 210 degrees. But now I am stuck.
Where do I go from here - or is this even the right way to try to solve this equation? Please help me.

start by graphing 5√3 - 5i

as (5√3, -5) in the argand plane
which would be in quadrant IV

r^2 = (5√3)^2 + (-5)^2 = 100
r = 10

we also know that tanØ = -5/(5√3) = -1/√3
( I know that tan 30° = tan π/6 = 1/√3)
but my angle is in IV, so Ø = 11π/6 or 330°

so 5√3 - 5i = 10(cos 11π/6 + i sin 11π/6)

check:
RS = 10( √3/2 + i(-1/2)
= 5√3 - 5i
= original complex number

so in summary
for a + bi

1. sketch (a,b) to see which quadrant you are in
2. evaluate r,
with r^2 = a^2 + b^2 ---> r = | √(a^2 + b^2)
3. from tan Ø = |b/a| find the acute angel Ø
- for I, Ø is that acute angle
- for II , Ø = π - acute angle
- for III , Ø = π + acute angle
- for IV, Ø = 2π - acute angle

a + bi = r( cosØ + isinØ)

use your calculator to verfiy your answer.

To express the complex number 5√3 - 5i in polar form, you need to find the magnitude (r) and the argument (θ).

Step 1: Find the magnitude (r)
The magnitude is the distance from the origin to the complex number, which can be found using the Pythagorean theorem.

r = √((5√3)^2 + (-5)^2)
= √(75 + 25)
= √100
= 10

So, the magnitude (r) is 10.

Step 2: Find the argument (θ)
The argument is the angle between the positive real axis and the line connecting the origin to the complex number. To find the argument, you can use trigonometric functions.

tan(θ) = (imaginary part) / (real part)
tan(θ) = -5 / (5√3)
θ = arctan(-5 / (5√3))
θ ≈ -60° or 300° in the standard unit circle

Step 3: Express the complex number in polar form
The polar form of a complex number is given by:
z = r(cos(θ) + i sin(θ))

Substituting the values we found, the complex number 5√3 - 5i in polar form is:
z = 10(cos(300°) + i sin(300°))

To determine the correct answer choice, convert the angle to radians:
300° = (300 × π) / 180 = 5π/3

So, the correct answer is d: 10(cos(5π/3) + i sin(5π/3)).

To express a complex number in polar form, you need to find its magnitude (r) and argument (θ). Here are the steps you can follow to find the polar form of 5√3 - 5i:

Step 1: Find the magnitude (r):
The magnitude of a complex number can be found using the formula:
r = √(a^2 + b^2)
where a is the real part and b is the imaginary part of the complex number.

In this case:
a = 5√3 (real part)
b = -5 (imaginary part)

r = √((5√3)^2 + (-5)^2)
= √(75 + 25)
= √100
= 10

Therefore, the magnitude of 5√3 - 5i is 10.

Step 2: Find the argument (θ):
The argument of a complex number can be found using the formula:
θ = arctan(b/a)
where b is the imaginary part and a is the real part of the complex number.

In this case:
a = 5√3 (real part)
b = -5 (imaginary part)

θ = arctan(-5 / 5√3)
= arctan(-1/√3)
= -π/6

Therefore, the argument (θ) of 5√3 - 5i is -π/6.

Step 3: Write the complex number in polar form:
The polar form of a complex number is given by:
z = r(cosθ + sinθi)

Substituting the values we found:
5√3 - 5i = 10 (cos(-π/6) + sin(-π/6)i)

Now, simplify the trigonometric expressions:
cos(-π/6) = cos(π/6) = √3/2
sin(-π/6) = -sin(π/6) = -1/2

Therefore, the polar form of 5√3 - 5i is:
5√3 - 5i = 10 (√3/2 - 1/2i)

Comparing this result with the answer choices provided:
a: 10(cos 11π/6 + i sin 11π/6)
b: 10(cos 11π/6 - i sin 11π/6)
c: 5 (cos 11π/6 + i sin 11π/6)
d: 10(cos 5π/3 + i sin 5π/3)

From the polar form we obtained, it matches with option B:
10(cos(11π/6) - i sin(11π/6))

Therefore, the correct answer is option B.