Today in Precalculus

• Go over homework• Need a calculator• Notes: Converting between Polar

and Rectangular Equations • Homework

Graphing Polar EquationsChange calculator mode to POL and radians

Type in r = 5cos(2θ) (use X,T,θ,N button for θ)

Zoom - standard

Graph

x

y

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

c. Rewrite secθ as and cscθ to

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.

1

cos1

sin

Example 1

Convert to rectangular, identify the type of equation and check the graph.

r = – 4secθ

r = – 4

rcosθ = – 4

x = -4

A vertical line

1

cos

Example 2Convert 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

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 12x – y = 5 (equation of a line, y-int. -5, slope 2)

2rcosθ – rsinθ = 5

r(2cosθ – sinθ ) = 55

2cos sinr

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

Homework

Pg. 540: 35-49odd