6 CHAPTER 1. COORDINATES AND VECTORS θ x y P • ℓ O r → Figure 1.5: Polar Coordinates direction of ℓ amounts to a (further) rotation by π radians, so the point with polar coordinates ( r,θ ) also has polar coordinates ( − r,θ + π ). In fact, while a given geometric point P has only one pair of rectangular coordinates ( x,y ), it has many pairs of polar coordinates. Given ( x,y ), r can be either solution (positive or negative) of the equation r 2 = x 2 + y 2 (1.4) which follows from a standard trigonometric identity. The angle by which the x-axis has been rotated to obtain ℓ determines θ only up to adding an even multiple of π : we will tend to measure the angle by a value of θ between 0 and 2 π or between − π and π , but any appropriate real value is allowed. Up to this ambiguity, though, we can try to ±nd
This is the end of the preview. Sign up
access the rest of the document.
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.