Polar Coordinates

Polar Coordinates Exercises

Polar Coordinate System

Mathematics 54–Elementary Analysis 2

Institute of Mathematics University of the Philippines-Diliman

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Polar Coordinates Exercises

Graphs in Polar Coordinates Special Curves in Polar Coordinates

The Polar Coordinate System

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Polar Coordinates Exercises

Graphs in Polar Coordinates Special Curves in Polar Coordinates

Polar Coordinates

Example Plot the following points:

1 2

A = (1, π/4) B = (2, −π/4)

3 4

C = (−2, π/6) D = (−3, −π/3)

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Polar Coordinates Exercises

Graphs in Polar Coordinates Special Curves in Polar Coordinates

Polar Coordinates

Example Plot the following points:

1 2

B = 2, − π = (2, 7π/4) 4

A = 1, π = (1, 9π/4) = (−1, 5π/4) 4

3 4

D = −3, − π = (3, 2π/3) 3

C = −2, π = (2, 7π/6) 6

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Polar Coordinates Exercises

Graphs in Polar Coordinates Special Curves in Polar Coordinates

Polar Coordinates

Conversion Equations Polar to Cartesian

1

Cartesian to Polar

1

x = r cos θ y = r sin θ

2

2

r 2 = x2 + y 2 y tan θ = x

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Polar Coordinates Exercises

Graphs in Polar Coordinates Special Curves in Polar Coordinates

Polar Coordinates

Example Determine the polar coordinates of the point having Cartesian coordinates (− 3, 1). Solution. Recall that r 2 = x2 + y 2 and tan θ = x . Thus, r 2 = − 3 + 12 tan θ = 1 − 3

2 y

=⇒ =⇒

r=2 θ= 5π 6

Hence, the polar coordinates are 2, 5π or −2, 11π . 6 6

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Polar Coordinates Exercises

Graphs in Polar Coordinates Special Curves in Polar Coordinates

Polar Coordinates

Example Determine the Cartesian coordinates of the point having polar coordinates −5, − π . 3 Solution. Recall that x = r cos θ and y = r sin θ. Thus, x = −5 cos − π = − 5 3 2 y = −5 sin − π = 3

5 3 2

Hence, the Cartesian coordinates are − 5 , 5 2 3 . 2

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Polar Coordinates Exercises

Graphs in Polar Coordinates Special Curves in Polar...