Today in Precalculus

1. Today in Precalculus • Go over homework • Need a calculator • Notes: Converting between Polar and Rectangular Equations • Homework

2. 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

3. Converting Equations - HINTS To convert from polar to rectangular: • If equation has sinθ or cosθ , multiply both sides by r. Then convert to x and/or y (x=rcosθ and y=rsinθ ) • Convert r2 to x2 + y2 • Rewrite secθ as and cscθ to • Complete the square if necessary. • (x – a)2 + (y – b)2 = r2 Equation of a circle with center (a,b) and radius r.

4. 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

5. 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

6. Converting Equations - HINTS To convert from rectangular to polar: • Multiply out any squared binomial terms like (x – 3)2 • Replace x with rcosθ and y with rsinθ • Replace x2 + y2 with r2 • solve for r (may need to factor)

7. Example 1 2x – y = 5 (equation of a line, y-int. -5, slope 2) 2rcosθ – rsinθ = 5 r(2cosθ – sinθ ) = 5

8. 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

9. Homework Pg. 540: 35-49odd