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 at = / 4 is approximately 0.892. 
Answer 5 
The area of one leaf is about 4.189 square units. 
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