Unit 2: Polar Coordinates
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
