Unit 2: Polar Coordinates

Lakukan tugas rumah & ujian kamu dengan baik sekarang menggunakan Quizwiz!

In General, the graphs of equations of the form r=xasintheta or r=xacostheta are circles with radius a centered at the points with polar coordinates _________ and _________ respectively.

(a,pi/2), (a,0)

Polar Coordinates

(r, theta)

Rectangular Coordinates

(x,y)

Polar to Rectangular Equations

*The goal is to change all the r's and theta's to x's and y's.* 1. Convert all trig functions into sin theta and cos theta if necessary. 2. Substitute: x=rcos(theta) y=rsin(theta) r=√x²+y² or r²=x²+y² 3. Sometimes multiplying both sides by r is helpful because then you can use r²=x²+y² 4. Simplify completely

Rectangular to Polar Equations

*The goal is to change all the x's and y's to r's and theta's.* 1. Substitute: x=rcos(theta) y=rsin(theta) 2. Solve for r (or theta)

6 Types of Graphs

1. Circle 2. Cardioid 3. Limacon 4. Rose 5. Lemniscate 6. Spiral

2 Types of Intersection Points

1. Simultaneous (at the same theta) 2. Non-simultaneous (usually the pole)

Lesson 4

Intro to Polar Graphing 1

Lesson 4.5

Intro to Polar Graphing 2

Lesson 3.5

More Polar Equations

Lesson 7

Simultaneous Polar Graphing

Completing the Square

ax²+bx+c=0 1. Subtract c from both sides. 2. Perform: (b/2)² and add this quantity to both sides. 3. Factor the perfect square trinomial.

Graph of r=n is a...

circle centered at the pole (0,0).

To find non-simultaneous intersection points...

graph both equations and determine where the graphs cross.

Graph of theta=n is a...

line.

r=asin/cos(ntheta) *n*

n=odd, n=petals n=even, 2n petals

Circle

r=+/-acostheta Center: (a,0) r=+/-asintheta Center: (0,a)

Spiral

r=+/-atheta

Cardioid

r=a+/-acostheta r=a+/-asintheta a>0 Passes through pole

Limacon

r=a+/-bcostheta r=a+/-bsintheta a>0, b>0 Without Inner Loop: a>b Does not pass through pole With Inner Loop: a<b Passes through pole twice

Rose

r=acos(ntheta) r=asin(ntheta) a=not 0 a=length of petals If n (not 0) is even... 2n petals If n (not +/- 1) is odd... n petals

Rectangular to Polar Coordinates

r=√x²+y² theta=y/x

Lemniscate

r^2=a^2cos2theta r^2=a^2sin2theta a=not 0

To find the simultaneous intersection points...

set each equation equal to each other and solve for theta.

Trig Identities

sin²theta+cos²theta=1 tan theta=sin theta/cos theta csc theta=1/sin theta sec theta=1/cos theta cot theta=1/tan theta=cos theta/sin theta

Polar To Rectangular Coordinates

x=rcos(theta) y=rsin(theta)

Equation of a Circle

x²+y²=r² (x-h)²+(y-k)²=r²

r=acos(ntheta)

Symmetrical over the x-axis/horizontal axis

r=asin(ntheta)

Symmetrical over the y-axis/vertical axis

r=asin/cos(ntheta) *a*

The further a is from 0, the greater the radius. The closer a is to 0, the smaller the radius. -a = reflection in the x-axis

Plotting Polar Coordinates

If r is positive, (r, theta) plotted on terminal side of theta. If r is negative, (r, theta) plotted on opposite ray of the terminal side. If theta is positive, (r, theta) plotted on terminal side of theta. If theta is negative, (r, theta) reflected over x-axis.

Lesson 8

More Polar Practice

Lesson 1

Polar Coordinates

Lesson 2

Polar Coordinates Practice

Lesson 3

Polar Equations

Lesson 5

Polar Graphing Exploration

Lesson 6

Polar Graphing Summary

Lesson 9

Polar Review


Set pelajaran terkait

Spinal Cord & Spinal Nerves: Anatomy Chapter 14

View Set

From the Middle Ages through the Renaissance

View Set

Texas Insurance License Ch.1 Quiz

View Set

kidney, pancreas, liver transplant

View Set

Food & Safety Taps practice test

View Set

Macbeth study questions (for act 1)

View Set