Pre-Calc Exam Notes 151

Pre-Calc Exam Notes 151 - in situations when there is...

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Polar Coordinates Section 6.4 151 Example 6.20 Prove that the distance d between two points ( r 1 , θ 1 ) and ( r 2 , θ 2 ) in polar coordinates is d = r r 2 1 + r 2 2 2 r 1 r 2 cos ( θ 1 θ 2 ) . (6.11) Solution: The idea here is to use the distance formula in Cartesian coordinates, then convert that to polar coordinates. So write x 1 = r 1 cos θ 1 y 1 = r 1 sin θ 1 x 2 = r 2 cos θ 2 y 2 = r 2 sin θ 2 . Then ( x 1 , y 1 ) and ( x 2 , y 2 ) are the Cartesian equivalents of ( r 1 , θ 1 ) and ( r 2 , θ 2 ), respectively. Thus, by the Cartesian coordinate distance formula, d 2 = ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 = ( r 1 cos θ 1 r 2 cos θ 2 ) 2 + ( r 1 sin θ 1 r 2 sin θ 2 ) 2 = r 2 1 cos 2 θ 1 2 r 1 r 2 cos θ 1 cos θ 2 + r 2 2 cos 2 θ 2 + r 2 1 sin 2 θ 1 2 r 1 r 2 sin θ 1 sin θ 2 + r 2 2 sin 2 θ 2 = r 2 1 (cos 2 θ 1 + sin 2 θ 1 ) + r 2 2 (cos 2 θ 2 + sin 2 θ 2 ) 2 r 1 r 2 (cos θ 1 cos θ 2 + sin θ 1 sin θ 2 ) d 2 = r 2 1 + r 2 2 2 r 1 r 2 cos ( θ 1 θ 2 ) , so the result follows by taking square roots of both sides. In Example 6.17 we saw that the equation x 2 + y 2 = 9 in Cartesian coordinates could be expressed as r = 3 in polar coordinates. This equation describes a circle centered at the origin, so the circle is symmetric about the origin. In general, polar coordinates are useful
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Unformatted text preview: in situations when there is symmetry about the origin (though there are other situations), which arise in many physical applications. Exercises For Exercises 1-5, convert the given point from polar coordinates to Cartesian coordinates. 1. (6,210 ◦ ) 2. ( − 4,3 π ) 3. (2,11 π /6) 4. (6,90 ◦ ) 5. ( − 1,405 ◦ ) For Exercises 6-10, convert the given point from Cartesian coordinates to polar coordinates. 6. (3,1) 7. ( − 1, − 3) 8. (0,2) 9. (4, − 2) 10. ( − 2,0) For Exercises 11-18, write the given equation in polar coordinates. 11. ( x − 3) 2 + y 2 = 9 12. y =− x 13. x 2 − y 2 = 1 14. 3 x 2 + 4 y 2 − 6 x = 9 15. Graph the function r = 1 + 2 cos θ in polar coordinates....
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