10.4 Polar Coordinates Miss Battaglia AP Calculus

Coordinate Conversion The polar coordinates (r, Θ ) of a point are related to the rectangular coordinates (x,y) of the point as follows. 1. x = rcos Θ 2. tan Θ =y/x y = rsin Θ r 2 = x 2 + y 2

Polar-to-Rectangular Conversion 1. Find the rectangular coordinates for the point (r, Θ )=(2, π ) 2. Find the rectangular coordinates for the point (r, Θ )=

Rectangular-to-Polar Conversion 1. Find the polar coordinates for the point (x,y)=(-1,1) 2. Find the polar coordinates for the point (x,y)=(0,2)

Graphing Polar Equations Describe the graph of each polar equation. Confirm each description by converting to a rectangular equation. a. r = 2b. Θ = π /3c. r=sec Θ

Slope in Polar Form If f is a differentiable function of Θ, then the slope of the tangent line to the graph of r=f( Θ ) at the point (r, Θ ) is provided that dx/d Θ ≠0 at (r, Θ ).

Finding Horizontal and Vertical Tangent Lines Find the horizontal and vertical tangents to the graph of r = 1 - sin Θ

Finding Horizontal and Vertical Tangent Lines Find the horizontal and vertical tangents to the graph of r = 3cos Θ Use the trig identities… cos 2 Θ – sin 2 Θ = cos2 Θ 2sin Θ cos Θ = sin2 Θ

Homework Read 12.3 and 12.4 Page 707 #2,5,6,7,9,29,35