Math 2414 Section 10 2 Polar Coordinates Defining Polar Coordinates Like Cartesian coordinates polar coordinates are used to locate points in the plane When working in polar coordinates the origin of the coordinate system is also called the pole and the r q The x axis is called the polar axis The polar coordinates for a point P have the form radial coordinate r describes the signed or directed distance from the origin to P The angular coordinate q describes an angle whose initial side is the positive x axis and whose terminal side lies on the ray passing through the origin and P Positive angles are measured counterclockwise from the positive x axis With polar coordinates points have more than one representation for two reasons First angles r q and r q 2p refer to are determined up to multiples of 2p radians so the coordinates the same point Second the radial coordinate may be negative which is interpreted as follows r q and r q are reflections of each other through the origin This means that The points r q r q p and r q p all refer to the same point The origin is specified as 0 q in polar coordinates where q is any angle 5p Q 1 4 in polar coordinates giving two alternative representations for the point Graph 7p R 1 Graph 4 in polar coordinates giving two alternative representations for the point 3p S 2 Graph 2 in polar coordinates giving two alternative representations for the point Converting between Cartesian and Polar Coordinates A point with polar coordinates r q has Cartesian coordinates x y where x r cos q and y r sin q A point with Cartesian coordinates x y 2 has polar coordinates 2 2 r x y and tan q r q where y x 3p P 2 Express the point with polar coordinates 4 in Cartesian coordinates Express the point with Cartesian coordinates Q 1 1 in polar coordinates Basic Curves in Polar Coordinates A curve in polar coordinates is the set of points that satisfy an equation in r and q The polar equation r 3 is satisfied by the set of points whose distance from the origin is 3 The angle q is arbitrary because it is not specified by the equation so the graph of r 3 is the circle of radius a 3 centered at the origin In general the equation r a describes a circle of radius centered at the origin p q 3 is satisfied by the points whose angle with respect to the positive x axis is The equation p 3 Because r is unspecified it is arbitrary and can be positive or negative Therefore q p 3 describes the line through the origin making an angle of p 3 with the positive x axis More generally positive x axis q q0 describes the line through the origin making an angle of q0 with the Convert the polar equation r 6 sin q to Cartesian coordinates and describe the corresponding graph Graph r 7 r 4 cos q and r 7 sin q on the same axis system Graphing in Polar Coordinates Conceptually the easiest graphing method is to choose several values of q calculate the corresponding r values and tabulate the coordinates The points are then plotted and connected with a smooth curve Graph the polar equation r f q 1 sin q r f q An alternative method for graphing polar curves is to graph as if r and q were Cartesian coordinates with q on the horizontal axis and r on the vertical axis Be sure to choose an interval in q on which the entire polar curve is produced Use the Cartesian graph as a guide r q on the final polar curve to sketch the points Use the previous method to graph the polar equation r 3sin 2q y x Graph r 5 5sin q y x Graph r 2 4 cos q y x r cos 2q 5 Consider the curve described by Give an interval in q that generates the entire curve and then compare with the graph of the curve 0 P r f q In general an interval over which the complete curve is guaranteed to be generated must satisfy two conditions P is the smallest positive number such that f 0 f P P is a multiple of the period of f so that and P is a multiple of 2p so that the points 0 f 0 and P f P are the same r a cos kq r a sin kq In general polar rose curves have equations of the form or where a 0 and k is a positive rational number The length of each petal is a The number of petals is determined by k If k is an even integer then there are 2k number of petals If k is an odd integer then there are k number of petals If k is not an integer then the number of petals is more r a sin kq complicated Below is a chart for polar roses given by the equation with a 1 and n k d where n and d are the first seven whole numbers