University:
University of California, Berkeley
Course:
MATH 51 | Calculus I
Academic year:

2020
Views:

336

Pages:

1
Author:

SpaceCadet101

Polar Coordinate Pair In a rectangular coordinate system, the ordered pair (a, b) denotes the point whose directed distances from the x− and y−axes are b and a, respectively. Another method for representing points is to use polar coordinates. To define p olar c oordinates, w e fi rst fix an ori gin O ( ca ll the pol e) and an ini tial ray from O. Usually the positive x-axis is chosen as the initial ray. Then the point P can be located by assigning to it a polar coordinate pair (r, θ) in which r gives the directed distance from O to P and θ gives the directed angle from the initial ray to ray OP . So we label the point P as P (r, θ). Exercise 1. Find all the polar coordinates of the point P (2, π/6). Exercise 2. Plot the points with the given polar coordinates. √ (a) (−3, 5π/6) (b) (5,tan−1 (4/3)) (c) (−1, 7π) (d) (2 3, 2π/3) Class √ Exercise 1. Plot the point whose polar coordinates are given. (b) (1, 5π/2) (c) (2, −7π/6) (a) (− 2, 5π/4) Exercise 3. Sketch the graph of the following polar equations: (a) r = 4 sin θ (b) r = 2 + 2 cos θ (c) r = 2 + 4 cos θ (d) r = a sin θ Class Exercise 2. Graph the polar curve. (a) r = 1 + cos θ (b) r = 2 − 2 cos θ (c) r2 = -sin 2θ (d) r = 1 - sin θ (e) r = 1 - 2 sin 3θ (f) r = sin(θ/2) (g) r = θ (h) r = 1 + sin θ (i) r = 2 cos 3θ (j) r = 1 + 2 sin θ Relationship between rectangular and polar coordinates: The rectangular coordinates (x, y) and polar coordinates (r, θ) of a point P are related as follows: (i) x = r cos θ, y = r sin θ (ii) r2 = x2 + y 2 , tan θ = y x if x ̸= 0 Exercise 4. Find the Cartesian coordinates for the following points that are given in polar coordinates: √ √ (a) ( 2, π/4) (b) (1, 0) (c) (0, π/2) (d) (− 2, π/4) Class Exercise 3. Find the Cartesian coordinates for the following points that are given in polar coordinates: √ (a) (−3, 5π/6) (b) (5,tan−1 (4/3)) (c) (−1, 7π) (d) (2 3, 2π/3) Exercise 5. Find an equation in x and y that has the same graph as the polar equation r = a sin θ, with a ̸= 0. Sketch the graph. Exercise 6. Replace the following polar equations by equivalent Cartesian equations and identify their graphs. 4 (a) r cos θ = -4 (b) r2 = 4r cos θ (c) r = 2cosθ−sinθ Class Exercise 4. Replace the polar equation by an equivalent Cartesian equation. Then identify or describe the graph. (a) r sin θ = 0 (b) r cos θ = 0 (c) r = 4 csc θ (d) r = -3 sec θ (e) r cos θ + r sin θ = 1 5 (f) r2 = 1 (g) r2 = 4r sin θ (h) r = sinθ−2cosθ (i) r2 sin 2θ = 2 (j) r = cot θ csc θ