day 4: polar coordinates · day 4 polar notes.notebook 1 ... (roses) r = a sin nθ and r = a cos...
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Day 4 polar Notes.notebook
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May 11, 2017
Day 4: Polar CoordinatesA coordinate system specifies the location of a point on a plane. Normally we use the Cartesian or rectangular coordinate system.
Polar Coordinate System:• Uses distances and directions to specify the locations of
points in the plane.• Starts with a fixed point O called the origin (or pole).• A ray is drawn from from O, called the polar axis (usually
the positive xaxis).• Any point P in the plane can be located by the length r of
the segment that connects O and P, and the angle θ between the polar axis and the segment: (r, θ)
We will use radians in polar coordinates.
Day 4 polar Notes.notebook
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θ will be positive if it is in the counterclockwise direction and negative if it is in the clockwise direction.
Pole = (0, θ)
P
Example: List all possible coordinates of the point P?
r is allowed to be negative. When r is negative, θ is the measure of any angle that has the ray opposite ray OP as its terminal side.
π/3
P2
π/3
P
2
2
(2, π/3) (2, π/3)4π/3
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Ex) Graph each of the following on polar graph paper:
a) P (1, 3π/4) b) P (3, π/6) c) P (3, 3π) d) P (4, π/4)
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To change from polar to rectangular coordinates:Remember: cos θ = x/r sinθ = y/rSolve for the formulas for x and y and you will get:X = rcosθ y = rsinθRectangular coordinates (rcosθ, rsinθ)
Ex) Change (4, 2π/3 ) from polar to rectangular coordinates.
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To change from rectangular to polar coordinates:To find r, remember that x2 + y2 = r2
To find θ, tan θ = y/xSince r can be positive or negative, there are two answers.
4) Change (2, 2) to polar coordinates
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Convert to polar form: To convert an equation from rectangular to polar form, you must rewrite the equation using r instead of x and y. Use the formulas
x = r cos θ and y = r sin θ
Ex) Write the rectangular equation (x 3)2 + y2 = 9 in polar form.
Ex) Write the rectangular equation x2 + (y 1)2 = 1 in polar form.
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Convert to Rectangular Form: To convert an equation from polar to rectangular form, you must use the formulas
r2 = x2 + y2 and tan θ = y/x
Ex) Write the polar equation r = 6 cos θ in rectangular form.
Ex) Write the polar equation r = 3 in rectangular form.
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Graphing Polar EquationsTwo methods can be used:1. Plot points and connect the points.2. Convert polar to rectangular form and see if it is a graph you recognize.
Ex) Graph polar equation r = 3
6) Graph π/3 and rewrite in rectangular form
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7) Graph r = 2sinθ and convert to rectangular form
8) Graph r = 2 + 2cosθ
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In general:Graphs with equations: (circles & spirals)
r = a circles with radius |a| and center at pole
r = a θ spirals
r = a sin θ and r = a cos θ are circles with radius |a|
center (a, π/2) center (a, 0)
Graphs with equations: (Roses)
r = a sin nθ and r = a cos nθ are roses (has petals)
if n is ODD (nleaved) if n is EVEN (2nleaved)
Graphs with equations: (Limacons)
r = a (1 ± sinθ ) and r = a (1 ± cosθ) are cardioids
(heart shaped)
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Limacons:
r = a ± b sinθ and r = a ± b cosθ
(a > 0, b > 0)
orientation depends on trig function sine or cosine and sign of b
a < b limacon with inner loop a > b dimpled limacon
a = b cardioid a ≥ 2b Convex limacon
more limacons: 1 + c sinθ and 1 + c cosθ
If c > 1, it has an inner loop (bigger c bigger loop!)
If c = 1, no loop
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Graphs with equations: (Lemniscates)
Figureeight shaped curves
r2 = a2 sin 2θ r2 = a2 cos 2θ