Precalc Prt2 Unit 1: Polar Coordinates and Complex Numbers

Simplify (3-3i) + (-3-6i)

-9i

Find the absolute value of the following complex number: |-6-8i|

10

Simplify the following complex number: i^5

i

x=rcosθ and y=rsinθ

polar to rectangular

Convert the rectangular equation, x = 7, into polar form.

r=7secθ

There are an infinite number of coordinates that can represent the same point in the polar plane.

true

Given the polar form of the following complex number, convert it into rectangular form: 2(cos4π/3 + isin4π/3)

−1−√3i

Raise 1+√3i to the 4th power and express the answer in rectangular form.

−8−8√3i

The equation, r=3r=3 , is written in _________ form

polar

Given the rectangular coordinates, (1,−√3), of a point, select all of the possible polar coordinates of that same point.

(2, 5π/3) (2,−π/3)

When calculating "θ" using the arctan function to write the complex number, −2+3i , in polar form, you must do what to find the actual "θ" that exists in the polar form?

Take 180 - θ

In the polar coordinate system, "r" is defined as

The distance from the pole to the point

When using DeMoivre's Theorem to evaluate either a complex number raised to a power or finding the roots of a complex number, the complex number you are evaluating must be in _____________ form

polar

Mr. DeMoivre is credited with discovering a useful pattern for _______________________ that became DeMoivre's Theorem.

evaluating powers of complex numbers

When finding the nth roots of a complex number, the exact values of these roots will be given in ______________ form, and the approximate values of these roots will be given in ________ form

polar, rectangular

If you were asked to find the fifth roots of −2−2i, what would be the values for the following:

r= NOT 2 0= NOT 45 degrees n= NOT 4 k= NOT 0-5

Choose the correct equation(s) for finding the nth roots of a complex number from the list below.

NOT r^n(cos(nθ)+isin(nθ))

Convert the polar coordinates (5, 3π/2) to rectangular coordinates.

(0, -5)

Convert the following polar coordinates to rectangular coordinates. (-3, 240°)

(1.5, 2.6)

How is a point on the polar plane represented?

(r, θ)

Given the following image of a point graphed in the polar plane, choose all of the ways that this point can be represented in polar coordinates. The coordinate is on the diagonal of 5π/6

(−6, 11π/6) (−6, −π/6) (6, 5π/6) (6,−7π/6)

Convert (−1, −1) into polar coordinates. Select the correct answer below.

(√2, 5π/4)

Simplify the following complex number: i^24

1

1. Coordinates of Point B 2. Coordinates of Point D 3. Coordinates of Point C 4. Coordinates of Point A

1. (8, 45°) 2. (1, 210°) 3. (3, 300°) 4. (9, 120°)

Given the complex number, 0 − 2i, how would it be written in polar form?

2(cos3π/2 + isin 3π/2)

Find the quotient of: 4(cos9π/4+isin9π/4) and 2(cos3π/2+isin3π/2)

2(cos3π/4+isin3π/4)

Evaluate (12i−5)^3 and leave your answer in polar form with the angle in degrees and all numbers rounded to the nearest whole number.

2197(cos(338°)+isin(338°))

Evaluate [8(cos(π3)+isin(π3))]6 and express the answer in rectangular form.

262144

How many types of symmetry can there be for polar equations and their graphs? State your numerical answer below.

3

Convert the following complex number number from rectangular form to polar form, and identify the following concepts: 3+3i What is r? What is 0? What is the polar form?

3 squared 2 NOT 90 degrees 3 squared 2 (cos 45 degrees + i sin 45 degrees)

How many types of limacons are there? State your numerical answer below.

4

What is the absolute value of the complex number, 3−4i ?

5

When finding the 5th roots of a complex number, the "n" value will be ______ and the "k" values will be _______

5, NOT 1-5

Find the product of: 3(cos π/2 + isin π/2) and 2(cos π/4 + isin π/4)

6(cos 3π/4 + isin 3π/4)

Find the product of: 2(cos60°+isin60°) and 4(cos150°+isin150°)

8(cos210°+isin210°)

When asked to find the eighth roots of 2(cos(π/4)+isin(π/4)), what is the polar form of the eighth root (the root with the largest k-value)?

8^√2(cos(57π/32) + isin (57π/32))

Given the rectangular form of the equation, (x−2)^2 + y^2 = 4 answer the following questions: This is a very common form of an equation. What would this equation look like graphically?: What is the polar form of this equation?:

A circle r = 4 cosθ

Given the rectangular form of the equation, y=(√3)(x) answer the following questions: This is a very common form of an equation. What would this equation look like graphically? What is the polar form of this equation?

A line NOT r = 3 squared

Given the polar form of the equation, r=6secθ answer the following questions: This is a very common form of an equation. What would this equation look like graphically? : What is the rectangular form of this equation?:

A line NOT y=6

Select the graph that represents the point (-4, -2 pi/3)

C

Given the following graph of a point on the polar plane, name another set of coordinates that could also represent this point using a positive "r" value and a negative "θ" value, where -360 ≤ θ ≤ 0. State your answer in the form: (r,θ) with "θ" in degrees without the degree symbol. (3, 45°)

NOT (1.5, 90°) or (1, -90°)

The complex conjugate of (4−8i) is:

NOT (−4+8i)

Given the complex number, 0 − 2i, how would it be written in polar form?

NOT 2(cos π/2 + isin π/2)

Find the quotient of: 10(cos45°+isin45°) and 2(cos15°+isin15°)

NOT 20(cos60°+isin60°)

In the polar coordinate system, the point (0, 0) is called the _______________, and the positive x-axis is called the _____________

Pole, polar axis

Evaluate: 5(cos135° + isin135°) · 2(cos45° + isin45°)

The rectangular form is: -10 The polar form is: 10(cos 180 degrees + i sin 180 degrees)

Evaluate: 6(cos135°+isin135°) ÷ 2(cos45°+isin45°)

The rectangular form is: 3i The polar form is: 3(cos 90 degrees + i sin 90 degrees)

What is the standard form of a complex number?

a + bi

Select all of the types of limacons from the choices below:

cardioid inner loop dimpled convex

The point with polar coordinates (−2, −135°) could also be expressed as (2, 225°).

false

When converting a complex number from rectangular form to polar form, "θ" must always be expressed in radians.

false

The absolute value of a complex number, a+bi, is represented graphically as the

length of the diagonal from the origin out to the point

In the polar form of a complex number, "r" is called the __________ and "θ" is called the _______

modulus, argument

A polar equation contains only ___________, and a rectangular equation contains only __________

r, 0 and x,y

Convert the rectangular equation, x=y^2 into polar form.

r=cotθcscθ

Convert the rectangular equation, y = -5, into polar form.

r=−5cscθ

A complex number consists of two parts, a _______ and a(n) _______________ part

real, imaginary

r= √x2+y2 and θ=tan^−1(y/x)

rectangular to polar

The polar coordinate system is _____ dimensional.

two

Convert the following polar equation into rectangular form: r = 1 / cosθ + sinθ

x + y = 1

When converting a polar equation into rectangular form, what are the conversion equations you can use? Select all that apply.

y = rsinθ x = rcosθ And one more

What are the four conversion equations used to convert between polar and rectangular equations?

y= r sin θ tan θ = y/x x = rcosθ r = √x^2+y^2

Convert the following polar equation into rectangular form: θ=π/4

y=x

If you have a complex number "z" and raise it to the 4th power, the correct equation to represent this is:

z^4 = r^4 (cos(4θ) + isin (4θ))

If you were asked to convert the complex number, -4 - 5i, into polar form, what values of " r " and " θ " would be in the polar form?

θ = NOT 51.3 r = NOT 33 squared