Today in Precalculus
• Go over homework
• Need a calculator
• Notes: Converting between Polar and
Rectangular Equations
• Homework
Graphing Polar Equations
Change calculator mode to POL and radians
Type in r = 5cos(2θ) (use X,T,θ,N button for θ)
Zoom - standard
Graph
y
x
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Converting Equations - HINTS
To convert from polar to rectangular:
a. If equation has sinθ or cosθ , multiply both sides by r.
Then convert to x and/or y (x=rcosθ and y=rsinθ )
b. Convert r2 to x2 + y2
1
c. Rewrite secθ as 1
and cscθ to
cos 
sin 
d. Complete the square if necessary.
e. (x – a)2 + (y – b)2 = r2
Equation of a circle with
center (a,b) and radius r.
Example 1
Convert to rectangular, identify the type of equation and
check the graph.
r = – 4secθ
1
r=–4
cos 
rcosθ = – 4
x = -4
A vertical line
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Example 2
Convert to rectangular, identify the equation, and check graph.
r = 2cosθ + 2sinθ
r2 = 2rcosθ + 2rsinθ
(multiply both sides by r)
x2 + y2 = 2x + 2y
x2 – 2x + y2 – 2y = 0
x2 – 2x + 1 + y2 – 2y + 1 = 1 + 1
(complete the square)
(x – 1)2 + (y – 1)2 = 2
Circle with center (1,1) and radius of 2
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Converting Equations - HINTS
To convert from rectangular to polar:
a. Multiply out any squared binomial terms like (x – 3)2
b. Replace x with rcosθ and y with rsinθ
c. Replace x2 + y2 with r2
d. solve for r (may need to factor)
Example 1
2x – y = 5
(equation of a line, y-int. -5, slope 2)
2rcosθ – rsinθ = 5
r(2cosθ – sinθ ) = 5
5
r
2 cos   sin 
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Example 2
(x – 2)2 + y2 = 4
(circle: center (2,0) radius 2)
x2 – 4x + 4 + y2 = 4
(multiply squared binomials)
x2 + y2 – 4x = 0
r2 – 4rcosθ = 0
(replace x2 + y2 with r2 and x with rcosθ)
r(r – 4cosθ ) = 0
(factor r)
r = 0 r – 4cosθ = 0
(set terms equal to zero)
r = 4cosθ
(solve for r)
r = 0 is a point at the pole
r=4cosθ is the equation
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Homework
Pg. 540: 35-49odd