624 CHAPTER 9 Polar Coordinates; Vectors In polar coordinates, the equation r 1, = whose graph is also the unit circle, does define a function. For each choice of ,θ there is only one corresponding value of r, that is, r 1. = Since many problems in calculus require the use of functions, the opportunity to express nonfunctions in rectangular coordinates as functions in polar coordinates becomes extremely useful. Note also that the vertical-line test for functions is valid only for equations in rectangular coordinates. Historical Feature Polar coordinates seem to have been invented by Jakob Bernoulli (1654—1705) in about 1691, although, as with most such ideas, earlier traces of the notion exist. Early users of calculus remained committed to rectangular coordinates, and polar coordinates did not become widely used until the early 1800s. Even then, it was mostly geometers who used them for describing odd curves. Finally, about the mid-1800s, applied mathematicians realized the tremendous simplification that polar coordinates make possible in the description of objects with circular or cylindrical symmetry. From then on, their use became widespread. Jakob Bernoulli (1654–1705) Concepts and Vocabulary 7. An equation whose variables are polar coordinates is called a(n) . 8. True or False The tests for symmetry in polar coordinates are always conclusive. 9. To test whether the graph of a polar equation may be symmetric with respect to the polar axis, replace θ by . 10. To test whether the graph of a polar equation may be symmetric with respect to the line 2 , θ π = replace θ by . 11. Rose curves are characterized by equations of the form r a n cos θ ( ) = or r a n a sin , 0. θ ( ) = ≠ If n 0 ≠ is even, the rose has petals; if n 1 ≠ ± is odd, the rose has petals. 12. True or False A cardioid passes through the pole. 13. Multiple Choice For a positive real number a, which polar equation is a circle with radius a and center a, 0 ( ) in rectangular coordinates? (a) r a2 sinθ = (b) r a2 sinθ = − (c) r a2 cosθ = (d) r a2 cosθ = − 14. Multiple Choice In polar coordinates, the points r, θ ( ) and r, θ ( ) − are symmetric with respect to which of the following? (a) the polar axis (or x -axis) (b) the pole (or origin) (c) the line 2 θ π = (or y -axis) (d) the line θ π = 4 ( ) = y x or 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Skill Building In Problems 15–30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. 15. r 4 = 16. r 2 = 17. 3 θ π = 18. 4 θ π = − 19. rsin 4 θ = 20. rcos 4 θ = 2 1. rcos 2 θ = − 22. rsin 2 θ = − 23. r 2cosθ = 24. r 2sinθ = 25. r 4sinθ = − 26. r 4cosθ = − 27. rsec 4 θ = 28. rcsc 8 θ = 29. rcsc 2 θ = − 30. rsec 4 θ = − ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 9.2 Assess Your Understanding 1. If the rectangular coordinates of a point are 4, 6 , ( ) − the point symmetric to it with respect to the origin is . (pp. 21–23) 2. The difference formula for cosine is A B cos( ) − = . (p. 511 ) 3. The standard equation of a circle with center at 2, 5 ( ) − and radius 3 is . (pp. 48–52) 4. Is the sine function even, odd, or neither? (pp. 422–423) 5. sin 5 4 π = (pp. 403–405) 6. cos 2 3 π = (pp. 403–405)
RkJQdWJsaXNoZXIy NjM5ODQ=