Question 6.3.2: Finding Other Polar Coordinates for a Given Point The point ...
Finding Other Polar Coordinates for a Given Point
The point (2, \frac{π}{3}) is plotted in Figure 6.23. Find another representation of this point in which
a. r is positive and 2π < θ < 4π.
b. r is negative and 0 < θ < 2π.
c. r is positive and -2π < θ < 0.
a. We want r > 0 and 2π < θ < 4π. Using (2, \frac{π}{3}), add 2π to the angle and do not change r.
(2, \frac{π}{3}) = (2, \frac{π}{3} + 2π) = (2, \frac{π}{3} + \frac{6π}{3}) = (2, \frac{7π}{3})
b. We want r < 0 and 0 < θ < 2π. Using (2, \frac{π}{3}), add π to the angle and replace r with -r.
(2, \frac{π}{3}) = (-2, \frac{π}{3} + π) = (-2, \frac{π}{3} + \frac{3π}{3}) = (-2, \frac{4π}{3})
c. We want r > 0 and -2π < θ < 0. Using (2, \frac{π}{3}), subtract 2π from the angle and do not change r.
(2, \frac{π}{3}) = (2, \frac{π}{3} – 2π) = (2, \frac{π}{3} – \frac{6π}{3}) = (2, -\frac{5π}{3})