Module 27 - Polar Functions

Module 27 - Answers

Lesson 1

Answer 1

27.1.1 The other possible coordinates are

, where k is any integer, or

, where k is any odd integer.

Answer 2

27.1.2

The polar point has approximate rectangular coordinates (1, 1.7321).

Lesson 2

Answer 1

27.2.1

The graph is a three leafed rose.

Answer 2

27.2.2

When n is even there are 2n leaves. When n is odd there are n leaves.

Answer 3

27.2.3

The graph has six overlapping leaves.

Answer 4

27.2.4

a = b gives a cardioid.

The graph shown is r = 2 + 2cos .

gives a limaçon with an inner loop.

The graph shown is r = 1 + 2 cos( ).

gives a dimpled limaçon.

The graph shown is r = 3 + 2 cos( ).

gives a convex limaçon.

Lesson 3

Answer 1

27.3.1 The area is approximately 12.5664 square units. Because the function r = 2 represents all points that are 2 units away from the origin, the graph is a circle of radius 2 units centered at the origin. Its exact area is A =

(2)² = 4

square units, which agrees with the area found.

Answer 2

27.3.2 The area inside the circle and outside the cardioid is

square units.

Self Test

Answer 1

The point (r, ) = (5.385, 1.951) in polar coordinates is the same as the rectangular point (-2, 5).

Answer 2

The point (x, y) = (1.618, 1.176) in rectangular coordinates is the same as the polar point

Answer 3

Answer 4

The slope of the tangent line to r = 2 - 3 cos

/ 4 is approximately 0.892.

Answer 5

The area of one leaf is about 4.189 square units.