unit 6

convert from polar to rectangular form: (2,90°) (-1,45°) (4,225°) (5,-π) (3, 5π/3)

(0,2) (-√2/2, -√2/2) (-2√2, -2√2) (-5,0) (3/2,-3√3/2)

if solving for θ in degrees: radians:

be in: degree mode radian mode

to convert from radians to degrees =

multiply by 180/π

convert from degrees to radians =

multiply by π/180

Change (-10, π/2) to rectangular form. Remember what a negative r value does!

(0,-10)

Convert the following polar coordinates into rectangular coordinates. 14. (2, 360°) 15. (-4, 60°) 16. (1.5, 80°)

(2,0) (-2,-2√3) (0.26,1.48)

convert each point given in rectangular coordinates to polar form: (-3,3) (0,-2) (-1/2, √3/2)

(3√2, 135°) (-2, 90°) (1,120°)

Ex 3. Multiply in rectangular form: (-2 + i)(3 + 4i) Ex 4. Convert the complex number into rectangular form: 4(cos210° + isin210°) Ex 5. Multiply in polar form: (8, 40°)(5, 25°) Ex 6. Divide in rectangular form: (1 - 2i)/(4 + i) Ex 7. Divide in polar form: (8, 40°)/(5, 25°)

-10-5i -2√2 - 2i (40,65°) (2-9i)/17 (8/5, 15°)

what is cos(-90)

0

what is sin(360)

0

Subtract (4 + 6i) - (3 - 5i)

1+11i

rewrite each equation in rectangular form: r = 7 10sinΘ = 5 Θ = π/3

x² + y² = 49 2y = √(x² + y²) y = (√3)x

sinθ =

y/r or y = rsinθ

tanθ =

y/x

Convert the following rectangular coordinates to polar coordinates. Give a second example of a polar coordinate that would have the same location. 1. (5,12) 2. (3, -3) 3. (0, -20) 4. (4, 5) 5. (2, 0) 6. (1, √3)

1. (13,67.38°) 2. (4.24,-45°) 3. (20,270°) 4. (6.40,51.34°) 5. (2,0°) 6. (2,60°)

convert from rectangular to polar form 1. (3,0) 2. (0,-8) 3. (-4,4) 4. (-1/2, √3/2) 5. (-√3/2, 1/2)

1. (3,0°) 2. (8,270°) 3. (4√2, 135°) 4. (1,120°) 5. (1,150°)

For each example, change the rectangular coordinate to polar form. 1. (0,5) 2. (2,-2) 3. (1,5) 4. (-10,0) 5. (4,6) 6. (-6,8)

1. (5,90°) 2. (2.83,-45°) 3. (5.10,78.69°) 4. (-10,180°) 5. (7.21,56.31°) 6. (10,126.87°)

the following are lines: 1. r = 6/sinθ 2. r = 3/cosθ what is the difference between #1 and #2? write a polar equation for the vertical line x = -5

1. y = 6 2. x = 3 #1 is a horizontal line at y = 6 #2 is a vertical line at x = 3 r = -5/cosθ or rcosθ = -5

multiply: (4+i)(2 - 3i) 3i(7 + 4i) (4 - 8i)(-6 + 10i) (7 + 5i)(20 - 2i) (6 - 3i)(2 + 4i) 8i(2 + 3i)

11-10i -12+21i 56+88i 150+86i 24+18i -24+16i

rewrite each equation in polar form: 11. x + y = 7 12. 4x^2 + y^2 = 9 13. y = 2x + 3

11. rcosθ = rsinθ = 7 12. r = 1.5 13. rsinθ = 2rcosθ + 3

Solve using your brain (and the complex number system). 12. x^2 = -1 13. x^2 = -25

12. i or -i 13. +/- 5i

(2+3i)(2-3i) 4i(2-3i (4-i)-5(2-3i) i^8+i^33 (1-3i)/(2+7i) (4+8i)/3i

13 12+8i -6+14i 1+i (-19-13i)/53 (-8+4i)/-3

Convert the following polar coordinates to rectangular coordinates. 13. (5, 45°) 14. (8, π) 15. (-3, -π) 16. (1,π/3 )17. (1, 3π2 )18. (2, 56°)

13. (5√2/2, 5√2/2) 14. (-8,0) 15. (3,0) 16. (1/2,√3/2) 17. (0,-1) 18. (1.12,1.66)

the following are all circles: find their radius and center: 5. r = 4sinθ 6. r = -4sinθ 7. r = 12cosθ 8. r = -12cosθ

5. r = 2, center (0,2) 6. r = 2, center (0,-2) 7. r = 6, center = (6,0) 8. r = 6, center = (-6,0)

We can think of complex numbers as numbers that have a real part and an imaginary part. As stated above, since imaginary is a bad name, let's think of complex numbers as...

2D numbers

360° =

2π radians

circumference equation

2πr

if the answer is a multiple of 60, then it has what number (usually) in the denominator

3

if the answer is a multiple of 30 but not 60, then it has what number (usually) in the denominator

6

add or subtract: (4+i)+(2 - 3i) 3i + (7 + 4i) + (2 - 7i) (4 - 8i) - (-6 + 10i) (7 + 5i) + (20 - 2i) (6 - 3i) - (2 + 4i) (8 - 4i) - (4 + i) + (2 + 3i)

6-2i 9 10-18i 27+3i 4-7i 6-2i

Now, try multiplying (3 + 3i)(-2 -4i) together both in rectangular form and polar form using the formula on the preceding page.

6-8i or (6√10, 288.4°) **double check to see if it converts

convert from polar to rectangular form 6. (-3,0°) 7. (5, 180°) 8. (6,45°) 9. (5.2,π) 10. (-1, 2π/3)

6. (-3,0) 7. (0,5) 8. (3√2,3√2) 9. (-5.2,0) 10. (1/2,-√3/2)

9. Write a polar equation for a circle with radius of 3 and with center at (3,0) 10. Write a polar equation for a circle with radius of 6 and center at (0,-6)

9. r = 6cosθ 10. r = -12sinθ

what is positive in the quadrant I? QII? QIII? QIV?

ALL (+,+) (aka (cosine is positive, sine is positive) sine and cosecant (-,+) tangent and cotangent (-,-) cosine and secant (+,-)

the 30s

30°, 150°, 210°, 330° all have the same sine and cosine and tangent except with negatives in different places they are closest to the horizontal axis

convert 3(cos45° + isin45°) into rectangular

3√2/2 + 3√2/2i

Common Polar Coordinate Graphs: Convert to rectangular form. Lines 4. θ = β 5. rcosθ = a 6. rsinθ = b

4. y = tanβ * x (this one is a linear equation with slope of tanβ(because than β is a constant)) 5. x = a (this one is a vertical line) 6. y = b (this one is a vertical line)

quadrantal angles

an angle in standard position whose terminal side coincides with one of the axes 0°, 90°, 180°, 270°, 360°

for the r(subscripts)#, then the number before signifies... the number before the theta signifies...

changes in size (amplitude) of petals number of pedals, and it also signifies: if odd, it gives you the # of pedals; if even, it doubles the # of pedals (ex: sin(2θ) = 4 pedals)

Note: Sometimes the converted equation comes out simpler or more complicated than the original equation. The benefit of being able to write an equation in 2 different systems (polar and rectangular) allows you to...

choose the simpler version of the equation to graph.

when graphing angles, you go in what direction

clockwise starting with positive horizontal axis

polar coordinates (r,θ) gives us a...

different way to represent a location.

first quadrant numerator&denominator rule with unit circle

π in numerator with 3, 4, or 6 in denominator

Imagine the number i. i =

√-1

convert r = -8cosθ into cartesian coordinates. use completing the square technique to transform the cartesian equation you find into something more recognizable

(x + 4)^2 + y^2 = 16 circle equation with center at (-4,0) radius of 4

convert -2 + 5i into polar

(√29)(cos111.8° + isin111.8°)

Now let's let i have some powers, just like the numbers we call real numbers. i^2 = i^3 = i^4 = i^5 = i^6 =

-1 √-1 1 √-1 -1

Convert into rectangular form of a complex number: (2, 135°)

-√2 + √2i

write each complex number in standard form (a + bi): a. 2(cos150 + isin150) b. 5(cosπ + isinπ) c. [3(cos30 + isin30)][5(cos60 + isin60)]

-√3 + i -5 15i

what is cos(150)

-√3/2

polar coordinates are not unique. there is more than...

1 coordinate that represents the same location

complex #s: imaginary and real part in...

1 equation ex: 1 + 1i

the theta that is the measure of the angle is also known as...

1 radian of measure

simplify and write in standard form a + bi: 1. i^2 2. i^99 3. -3i + 5i 4. (1 + 6i) + (3 - 5i) 5. (4 + 2i) - (-1 + 3i) 6. 5(2i) 7. (4i)(3i) 8. 2(10 + i) 9.(5 + i)(5 - i) 10. 3i/i^2 11. (1+3i)/(2-4i) 12. (6+2i) to polar 13. -3(cos30+isin30) to rectangular

1. -1 2. -i 3. 2i 4. 4+i 5. 5-i 6. 10i 7. -12 8. 20+2i 9. 26 10. -3i 11. (-10+10i)/20 12. √40 (cos18.43+isin19.43) 13. -3√3/2 + -3/2i

convert the following degrees into radians. leave solutions rounded to 2 decimal places. 1. 13.9° 2. -27° 3. 82°30'30"

1. 0.24 radians 2. -0.47 radians 3. 1.44 radians

convert the following radians into degrees. round solutions to 2 decimals places when necessary. 1. 7π/6 2. 3π/8 3. 8.5

1. 210° 2. 67.5° 3. 487.01°

convert 2x - 5x^3 = xy into polar coordinates

2 - 5x^2 = y or 2rcosθ - 5r^3cos^3θ = rcosθ * rsinθ

the 60s

60°, 120°, 240°, 300° all have the same sine and cosine and tangent except with negatives in different places they are closest to the vertical axis

Multiply (4 - 3i)(1 + i) Divide (3+i)/(1-2i) Multiply (20, π/3)(5, π/2) Multiply (10, π/4)(2, π/6) Divide (20, π/3)/(5, π/2) Divide(4, π/2)/(2, π/4) Convert into polar form: 9 + 3i Convert into polar form: -2 - i Convert into polar form: 5i Convert into rectangular form of a complex number: (4, 90°) Convert into rectangular form of a complex number: 2(cos270° + isin270°) Convert into rectangular form of a complex number: -3(cosπ/2 + isinπ/2)

7+i (1+7i)/5 (100, 5π/6) (20,5π/12) (4,-π/6) (2,π/4) 3√10(cos18.4°+isin18.4°) √5(cos206.6°+isin206.6°) 5(cos90°+isin90°) 4i -2i -3i

number system that i have learned

Complex Numbers -Real Numbers --Rational Numbers ---Integers

Multiplying and Dividing Complex Numbers (Polar Form)

Multiplication and Division of complex numbers in polar form is often quicker than rectangular form! Here are the rules you can apply for complex numbers written in polar form: Multiplying : Multiply the r values together and add the measures of the angles. (r1,Θ1)(r2,Θ2) = r1*r2(Θ1+Θ2) Ex: (2,30°)(5,60°) = (10, 90°) (-4,π)(3,π/4) = (-12, 5π/4) = (12, π/4) Dividing: Divide the r values and subtract the measures of the angles. (r1 ,Θ1)/(r2 ,Θ2) = r1/r2(Θ1+Θ2) Ex: (6, 90°)/(3, 30°) = (2,60°) (2,π/2)/(4, π) = (0.5,- π/2)

It is intuitively obvious to the most casual observer that 4 is a complex number. 4 = 4 + 0i

Similarly, any imaginary number is also a complex number. 3i = 0 + 3i

SOH CAH TOA stands for?

Sine: O/H Cosine: A/H Tangent: O/A

Addition of Complex Numbers:

To add complex numbers, simply add the real parts separately from the imaginary parts. When two or more complex numbers are added together, the result is a complex number. In symbols: (a + bi) + (c + di) = (a + c) + (b + d)I

A complex number can be written in the form a + bi...

a is the real part and bi is the imaginary part. a is the horizontal component and b is the vertical component.

A winter hiker was resting on a mountain peak listed on his mathematical G.P.S. of 4 - 12i. The hiker would like to return to a comfortable lodge with a cozy fire with a G.P.S. reading of 6 + 2i. a. What is the difference between the hiker's current location and the lodge? Answer as a complex number. How far is the hiker from the lodge? c. In what direction should the hiker travel?

a. -2-14i b. 14.1 units c. 82 degrees

the angle θ indicates the...

direction one would travel from the origin to get the point θ is the angle that the ray from the origin to the point makes with the positive x-axis. the angle is measured counterclockwise from the positive x-axis.

the value of r indicates the...

distance that the point lies from the origin

Symmetry with respect to the y-axis is called the

line θ = π/2 this means that the shape is symmetric when flipped over the y-axis

fourth quadrant numerator&denominator rule with unit circle

numerator is (2 x denominator) - 1

Symmetry can be helpful when polar equations get more complicated and a table of values becomes necessary. By checking for symmetry, it may be possible to...

reduce the number of points needed to draw the graph by hand. The following rules can be used to prove whether a polar graph has symmetry.

y =

rsinθ

tan =

sin/cos

standard position

start on horizontal axis (right side)

the points (-1,π) and (2, 3π/2) are given in polar coordinates. plot the points on the rectangular grid below

(1,0) (0,-2)

sine and cosine of the quadrantal angles 0&360: 90: 180: 270:

(1,0) (cosine is 1, sine is 0) (0,1) (-1,0) (0,-1)

convert to polar form then multiply: (1/2 + √3/2i)(√2/2 + √2/2i)

(1,60°)(1,45°) (1,105°)

Change (2, 30°) to rectangular form.

(1.73,1)

multiply or divide. if question is given in polar form, leave solution in polar form. if question is given in rectangular form, leave solution in rectangular form. a. (6,π/3)/(2,π/12) b. (3+i)/(1-2i) c. (-3 + i)(3 + 2i)

(3,π/4) (1+7i)/5 -11-3i

convert from rectangular to polar form: (3,-3) (-2,0) (0,2) (√3/2,1/2) (-√3/2, 1/2) (1/2, -√3/2)

(3√2,-45°) (2,π) (2,π/2) (1,π/6) (1,7π/6) (1,5π/3)

convert to polar form and divide: (2 + 2√3i) / (√3 + i)

(4, 60°) / (2, 30°) (2,30°)

zoom # to look like perfect circles

5

x =

rcosθ

how can the rectangular coordinate (-3,4) be converted into polar form?

this is in QII hypotenuse is 5 use inverse tan(4/3) to find theta = 53.13° the angel is not 53.13° because it is in QII and this is just the reference angle. it should start at the positive x-axis and go counterclockwise, therefore the angle is 126.87°

solve using the complex number system: x^2 = -16 x^2 + 5 = -11

x = +/-4i x = +/-4i

covert r = 2secθ to rectangular form

x = 2

convert each point given in polar coordinates to rectangular form: (5,45°) (3,π/6) (-3,-π/2)

(5√2/2, 5√2/2) (3√3/2, 3/2) (0,3)

divide: (4-7i)/3i (2+4i)/5i (2-i)/2i (10+5i)/i (1-5i)/(2+3i) (2+4i)/(3-5i) (3+7i)/(1-2i) (8+6i)/(2+4i)

(7+4i)/-3 (-4+2i)/-5 (1+2i)/-2 5-10i -1-i (-7+11i)/17 (-11+13i)/5 2-i

fill in ( , ) with sine, cosine, or tangent

(cosine, sine)

write each complex number in polar form. write each angle measure in degrees. -1 - i -√3 + i 2012

(√2, 225°) (2, 150°) (2012, 0°)

Add (-3 + i) + (2 + 3i)

-1+4i

angle θ in degrees, angle θ in radians, sin θ, cos θ, tan θ: 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360

0, 0, 1, 0 π/6, 1/2, (√3)/2, (√3)/3 π/4, (√2)/2, (√2)/2, 1 π/3, (√3)/2, 1/2, √3 π/2, 1, 0, undefined 2π/3, (√3)/2, -1/2, -√3 3π/4, (√2)/2, -(√2)/2, -1 5π/6, 1/2, -(√3)/2, -(√3)/3 π, 0, -1, 0 7π/6, -1/2, -(√3)/2, (√3)/3 5π/4, -(√2)/2, -(√2)/2, 1 4π/3, -(√3)/2, -1/2, √3 3π/2, -1, 0, undefined 5π/3, -(√3)/2, 1/2, -√3 7π/4, -(√2)/2, (√2)/2, -1 11π/6, -1/2, (√3)/2, -(√3)/3 2π, 0, 1, 0

convert the following degrees into radians. leave solution as reduced fractions. 1. 135° 2. -270° 3. 315°

1. 3π/4 radians 2. 3π/2 radians 3. 7π/4 radians

rewrite each equation in rectangular form 14. r = 12 15. rsinθ = 1/4 16. θ = 1/4

14. x^2 + y^2 = 144 15. y = 1/4 16. y = tan(1/4)(x)

csc = cot = sec =

H/O A/H H/A

Multiplication of Complex Numbers

To multiply two complex numbers, use either FOIL or the box method. Notice that we end up with only two terms. When two or more complex numbers are multiplied together, the result is a complex number. Ex: (2 + 3i)(-1 + 4i) = ?

Let's construct a triangle with the first side along the x-axis, the second side parallel to the y-axis and the third (terminal) side connecting the origin to the vertex (x,y). Now let's call the angle with its initial side along the positive x-axis and its terminal side connecting (0,0) to (x,y) . See diagram.

cover, do, check, correct Converting from polar to rectangular coordinates.

plot and label the following points 1. A(3,30°) B(1,210°) 2. C(2,45°) D(5,300°) 3. A(-2,100°) B(2,100°) 4 C(4,-180°) D(-2,180°)

cover, do, check, correct Notes and Guided Practice on Polar Coordinates

look at the light heart graphs of polar systems

do, cover, check, correct

there are + and - sides to the...

imaginary and real axeses

i is the...

imaginary dimension, and multiplying by i is what puts you in that imaginary dimension (look at the graph draw on this notes sheet)

imaginary numbers

in the second dimension negative = opposite x^2 = 9 and x^2 = -9

you can graph circles on the polar coordinate plane by using the radius to graph that level of the circle. you can also graph vertical and horizontal lines by the rings on the circle and what number the line is

look on Plotting Polar Equations by Hand and Symmetry in order to see this (both pages) LOOK ON THIS PAGE FOR EVERYTHING ACCORDING TO GRAPHING POINTS, LINES, AND CIRCLES!!!! cover, do, check, correct

polar equations can be...

plotted by hand. similarly to rectangular equations, polar equations can be graphed by making a table of values and then plotting the coordinates. remember it is important to determine how many degrees each "spoke" of the polar graph paper represents.

modes for polar systems

polar mode radian mode

symmetry with respect to the origin is called the

pole this means that the shape is symmetric when flipped over the origin from QI to QIII (aka the line of symmetry is a negative correlation (top left to bottom right))

write the equation x^2 +y^2 = 9 in polar form

r = 3

rewrite each equation polar form: x² + y² = 16 x + y = 16 y = √(25 - x²)

r = 4 rcosΘ + rsinΘ = 16 r = 5

from center to edge of circle

radius

rewrite each equation in polar form: x = 0 4x^2 + 4y^2 = 9 y = x - 5

rcosΘ = 0 r = 3/2 rsinΘ = rcosΘ - 5

complex #s: standard form in rectangular and polar

rectangular: a + bi polar: r(cosθ + isinθ)

for the heart drawings:

the number in the front most is being subtracted from or added to (ex: first 3: r = 3-3sinθ), it signifies the radius the minus sine in this example signifies if the heart is upside down or rightside up (upside down = +, rightside up = -) the second 3 in this example signifies the size of the heart. the crest at the top is the same (where the heart meets), but the rest of the heart is bigger if you increase? this number

symmetry with respect to the x-axis is called

the polar axis this means that the shape is symmetric when flipped over the x-axis

r^2 =

x^2 + y^2

rewrite each equation in rectangular form: r = 2 rsinΘ = 7 Θ = π/4

x^2 + y^2 = 4 y = 7 y = 1x

covert r = 3 into cartesian coordinates

x^2 + y^2 = 9 circle equation with center at origin and radius of 3

look at graph for i-Rithmetic (coordinate plane)

yuh

θ√°

yuh

unit circle

yuh.

θπ°

yuh.

180° =

π radians

Change the number (1 + i) into polar form.

√2(cos45°+isin45°)

What is sin45?

√2/2

equation: a + bi size of a + bi = size of (-x) =

√a^2 + b^2 √(-x)^2 = IxI

r =

√x^2 + y^2

when squaring, cubing, etc something with theta in it,...

don't put the exponent after the theta, put it before the theta on the tan, cos, sin, etc.

real dimension is on the _______ axis imaginary dimension is on the _________ axis

horizontal vertical

In the history of mathematics, some mathematicians have invented numbers to find creative solutions to problems. The number...

i is one of these important numbers.

converting from rectangular to polar coordinates

r is the distance that a point lies from the origin. r can be found from the coordinates (x,y) using the pythagorean theorem. instead of a^2 + b^2 = c^2, we use the variables x^2 + y^2 = r^2 θ is the angle that the ray connecting the origin to the point makes with the positive x-axis. θ can be found using the tangent ratio(and using inverse tangent)

Any real number is also a...

complex number

hand trick

cosine on right sine on left always in the square root, and always over 2

look at Common Polar Coordinate Graphs: Convert to rectangular form for #7

cover, do, check, correct

look at Guided Practice on the back page of Converting from polar to rectangular coordinates to graph the coordinates

cover, do, check, correct

look at graphs on part two of team test

cover, do, check, correct

how to plot polar coordinates

first determine how many degrees each "spoke" represents if a coordinate does not exactly fall on a gridline, estimate its location *a negative radius means that the terminal side goes in the opposite direction *the x-axis is referred to as the "polar axis" and y-axis is the "line θ = π/2"

In order to use polar form to help us with complex numbers, we need to be able to convert complex numbers from rectangular form into polar form... To convert into polar form:

Real + Imaginary = x + yi = rcosθ + (rsinθ)i = r(cosθ + isinθ) look at the plots on the page to compare the two types of graphing and switching between the two forms

Division of Complex Numbers (The denominator cannot be 0).

To divide one complex number by another, multiply the numerator and denominator by the conjugate of the denominator. The conjugate is the same first term and opposite second term. When one complex number is divided by another complex number (other than 0), the result is a complex number. The conjugate of a + bi is a - bi. To divide (a+bi)/(c+di), multiply by the conjugate of the denominator (a+bi)/(c+di)*(c-di)/(c-di)

Subtraction of Complex Numbers

To subtract one complex number from another, simply subtract the real parts (in proper order) separately from the imaginary parts (in proper order). When a complex number is subtracted from another complex number, the result is a complex number. In symbols: (a + bi) - (c + di) = (a - c) + (b - d)I

when r(subscript)# = something, if it has sin θ in the denominator, then... if it has cos θ in the denominator, then...

they are horizontal lines they are vertical lines

cosine = sine = tangent =

x coordinate y coordinate y coordinate/x coordinate

evaluate the following trig functions. simplify answers completely and leave all answers as exact radicals that have been rationalized. 1. cos(π/6) = 2. cos(5π/3) = 3. cos(5π/6) = 4. sin(8π/3) = 5. sin(5π/4) = 6. sin(7π/6) = 7. sin(2π) = 8. sin(-π/2) = 9. sin(7π) = 10. cos(5π/2) = 11. cos(0) = 12. cos(22π) =

1. (√3)/2 2. 1/2 3. -(√3)/2 4. (√3)/2 5. -(√2)/2 6. -1/2 7. 0 8. -1 9. 0 10. -1 11. 1 12. 1

look at the page with the table and graphs that has a paragraph at the beginning that starts with "symmetry can be..." and then the two pages that follow it (including the guided practice with the blank page on the back)

cover, do, check, correct

plot and label the points on the team test

cover, do, check, correct

***Note: Much of calculus requires FUNCTIONS. While x^2 + y^2 = 1 is not a function in rectangular coordinates because it...

fails the vertical line test, r = 1 is a function in polar coordinates because for each θ, there is only 1 r value.***

can you do 2 multiplications to equal a negative #:

i^2 = -1 this is a rotation of 2 a rotation from 1 to -1 (starting at the right horizontal axis) top rotation = over bottom rotation = under

Cartesian (or rectangular) coordinates, (x,y) give us one ways to...

identify a location of a point in a plane.

second quadrant numerator&denominator rule with unit circle

numerator is 1 less than the denominator

third quadrant numerator&denominator rule with unit circle

numerator is 1 more than the denominator

the point (3,3) is unique. that is to say, there is only...

one point in an x-y plane that has the coordinates x = 3 and y = 3 so if you asked to travel to the location (3,3) there is only one place it could be

radian definition

one radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of the circl

look at Plotting Polar Equations by Hand and Symmetry for A and B for the spoke and degrees of each spoke

cover, do, check, correct

cosθ =

x/r or x = rcosθ