842 CHAPTER 8 Applications of Trigonometry Graphs of Polar Equations An equation in the variables x and y is a rectangular (Cartesian) equation. An equation in which r and u are the variables instead of x and y is a polar equation. r = 3 sin u, r = 2 + cos u, r = u Polar equations Although the rectangular forms of lines and circles are the ones most often encountered, they can also be defined in terms of polar coordinates. The polar equation of the line ax + by = c can be derived as follows. Line: ax + by = c Rectangular equation of a line a1r cos u2 + b1r sin u2 = c Convert to polar coordinates. r 1a cos u + b sin u2 = c Factor out r. r = c a cos U +b sin U Polar equation of a line For the circle x2 + y2 = a2, the polar equation can be found in a similar manner. Circle: x2 + y2 = a2 Rectangular equation of a circle r2 = a2 x2 + y2 = r2 r =a or r = −a We use these forms in the next example. This is the polar equation of ax + by = c. These are polar equations of x2 + y2 = a2. Polar equation of a circle; r can be negative in polar coordinates. x y 0 3 –3 –3 3 y = x – 3 (rectangular) r = (polar) cos U – sin U 3 (a) Figure 64 −10 −10 10 10 Polar graphing mode (b) EXAMPLE 3 Finding Polar Equations of Lines and Circles For each rectangular equation, give the equivalent polar equation and sketch its graph. (a) y = x - 3 (b) x2 + y2 = 4 SOLUTION (a) This is the equation of a line. y = x - 3 x - y = 3 Write in standard form ax + by = c. r cos u - r sin u = 3 Substitute for x and y. r 1cos u - sin u2 = 3 Factor out r. r = 3 cos u - sin u Divide by cos u - sin u. A traditional graph is shown in Figure 64(a), and a calculator graph is shown in Figure 64(b). (b) The graph of x2 + y2 = 4 is a circle with center at the origin and radius 2. x2 + y2 = 4 r2 = 4 x2 + y2 = r2 r = 2 or r = -2 The graphs of r = 2 and r = -2 coincide. See Figure 65 on the next page. In polar coordinates, we may have r 60.
RkJQdWJsaXNoZXIy NjM5ODQ=