9-1: Polar Coordinates & 9-2: Graphs of Polar Equations

Objectives Graph points in polar coordinates. Graph simple polar equations. Determine the distance between two points with polar coordinates. Graph polar equations.

Real World Application Before large road construction projects, or even the construction of a new home, take place, a surveyor maps out characteristics of the land. A surveyor uses a device called a theodolite to measure angles. The precise locations of various land features are determined using distances and the angles measured with the theodolite.

Polar Graph

Polar Vocabulary Point O: pole or origin Polar axis: usually the horizontal ray directed right from the pole Point P is identified by polar coordinates in the form (r, θ). |r|: distance from the pole to P Θ: measure of the angle formed by OP and the polar axis (can be measured in degrees or radians)

r Values P(r,θ) P(r,θ) when r>0 P(r,θ) when r<0 θ θ P(r,θ)

Example Graph each point. A(-4,0) B(2,3π/2)

Example Graph the point. C(-2,-240°)

Uniqueness Polar coordinates are not unique. Every point can be represented by infinitely many pairs of polar coordinates by adding or subtracting 360° or 2π radians.

Example How else can you write (2, 120°) using degrees? (2, -240°) (2, 480°) (-2,-60°) (-2, 300°) (-2,-420°)

Example Name four different pairs of polar coordinates that represent point R on the graph with the restriction that -360°≤r≤ 360°. (2, 210°) (2, -150°) (-2, 30°) (-2, -330°) R

Polar Equation An equation expressed in terms of polar coordinates is called a polar equation.

Example Graph the polar equation r=6.

Example Graph the polar equation θ=5π/6.

Distance Formula in Polar Plane If and are two points in the polar plane, then Law of Cosines r2 r1

Surveying Example If one landmark is 450 feet away and 30° to the left and another landmark is 600 feet away and 50° to the right. Find the distance between the two landmarks. 450 ft 600 ft

9-2 Graphs of Polar Equations

Example 1 (p.561)

Example 2a (p. 562)

Example 2b (p.563)

Example 3 (p. 563)

------------------------------------------- Homework 9-1 p. 558 #6-13 all, #17-39 odds 2nd day: ------------------------------------------- p. 558: #14, 41, 43, 45 9-2 p. 565 #11-15 odd, 21