Pre-Calc Exam Notes 151

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

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.

Ask a homework question - tutors are online