1.) determine two pairs of polar coordinates for the point (-4sqrt3, 4sqrt3)with 0degrees less than or equal to theta less than or equal to 360degrees

2.) Use DeMoivre's Theorem to find (1-i)^10. write your answer in the form a+bi.

3.) Find the cube roots of -8. Write your answer in form a+bi

start with 3)

3) -8=8@180

roots: 2@60 and at each additional 120: 60, 180, 300

lets do another just in case you need practice: fifth root of 32@55 deg

on, 1/5 of 55 is 11. 1/5 of 360 is 72 so angles will be 11, then each 72 degrees for four additional roots. Notice the symettry when you sketch.

now, the magnitude fifth root of 55 is 2.228

2) 1-i= sqrt2@-45= sqrt2 @315

take that to the tenth power: (sqrt2)^10 = 2^5=32

angle? 315x10=3150
lets do the negative angle -45. that times 10 is -450 or -90 = 270 Nice.

then the answer is -32i

check that.
1) in polar, magnitude is sqrt(16*3*2)=9.80
angle is 135

check my work,I did most of this in my head with a migraine.

1. x^2 + y^2 = r^2 = 16*3+16*3

= 16*6
so r = +/- 4 sqrt 6
theta = 3 pi/4 or -pi/4

2. r = sqrt(1+1) = sqrt 2

theta = tan^-1 (1) = -pi/4
so
r e^i theta = sqrt 2 e^(-ipi/4 )

sqrt 2^10 = 2^5 = 32

-10 pi/4 = -5/2 pi = -pi /2 or -90

32 (cos -90 + i sin -90)

-32 i

thank you guys so much! I had a really hard time with these. I appreciate it :)

1.) To find the polar coordinates of a point, we need to determine the distance from the origin (r) and the angle (θ) it makes with the positive x-axis.

Given the point (-4√3, 4√3), we can calculate r using the distance formula:
r = √((-4√3)^2 + (4√3)^2) = √(48 + 48) = √96 = 4√6

Since the point lies in the second quadrant (negative x and positive y), the angle θ can be found using the inverse tangent function:
θ = tan^(-1)((4√3)/(-4√3)) = tan^(-1)(-1) = -45°

Since we want the angle to be between 0° and 360°, we can add 360° to -45° to get 315°:
θ₁ = -45° + 360° = 315°

Another valid angle in the given range can be obtained by adding 180° to θ₁:
θ₂ = 315° + 180° = 495°

Therefore, the two pairs of polar coordinates for the point (-4√3, 4√3) are (4√6, 315°) and (4√6, 495°).

2.) DeMoivre's Theorem states that for any complex number z = a + bi, where a and b are real numbers, and n is a positive integer:

z^n = (r^n) * [cos(nθ) + i sin(nθ)]

In this case, we have z = 1 - i and we want to find (1 - i)^10.

First, we need to find r and θ for z = 1 - i:
r = √(1^2 + (-1)^2) = √(1 + 1) = √2
θ = tan^(-1)((-1)/1) = -π/4 (in radians) or -45° (in degrees)

Now we can use DeMoivre's Theorem:
(1 - i)^10 = (√2^10) * [cos(10 * (-π/4)) + i sin(10 * (-π/4))]

Simplifying further:
(1 - i)^10 = 2^5 * [cos(-5π/2) + i sin(-5π/2)]

Since cos(-5π/2) = 0 and sin(-5π/2) = -1, we have:
(1 - i)^10 = 2^5 * [0 - i] = 32i

Therefore, (1 - i)^10 = 32i.

3.) to find the cube roots of -8, we can express -8 in polar form and then apply the formula for finding the cube roots of a complex number.

In polar form, -8 can be represented as 8 * [cos(π) + i sin(π)], since cos(π) = -1 and sin(π) = 0.

Now, let's find the cube roots of 8:
∛8 = 2 * [cos(0) + i sin(0)]

To find the other two complex cube roots, we need to divide the total angle of 360° into three equal parts. Therefore:
∛8 = 2 * [cos(0) + i sin(0)]
∛8 = 2 * [cos(120°) + i sin(120°)]
∛8 = 2 * [cos(240°) + i sin(240°)]

So, the three cube roots of -8 are 2, 2 * [cos(120°) + i sin(120°)], and 2 * [cos(240°) + i sin(240°)].
In standard form, they can be written as 2, -1 + √3i, and -1 - √3i, respectively.