Precalc Prt2 Unit 1: Polar Coordinates and Complex Numbers

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

-9i

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°)

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

10

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

Simplify the following complex number: i^5

i

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

x=rcosθ and y=rsinθ

polar to rectangular

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

r=7secθ

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

r=cotθcscθ

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

r=−5cscθ

r= √x2+y2 and θ=tan^−1(y/x)

rectangular to polar

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

true

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

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

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

real, imaginary

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

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

r, 0 and x,y

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))