Finding Other Polar Coordinates for a Given Point The point (2, π/3) is plotted in Figure 6.23. Find another representation of this point in which a. r is positive and 2π

Question

Finding Other Polar Coordinates for a Given Point

The point [latex](2, \frac{π}{3})[/latex] 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.

Accepted Answer

a. We want r > 0 and 2π < θ < 4π. Using [latex](2, \frac{π}{3})[/latex], add 2π to the angle and do not change r.

[latex](2, \frac{π}{3}) = (2, \frac{π}{3} + 2π) = (2, \frac{π}{3} + \frac{6π}{3}) = (2, \frac{7π}{3})[/latex]

b. We want r < 0 and 0 < θ < 2π. Using [latex](2, \frac{π}{3})[/latex], add π to the angle and replace r with -r.

[latex](2, \frac{π}{3}) = (-2, \frac{π}{3} + π) = (-2, \frac{π}{3} + \frac{3π}{3}) = (-2, \frac{4π}{3})[/latex]

c. We want r > 0 and -2π < θ < 0. Using [latex](2, \frac{π}{3})[/latex], subtract 2π from the angle and do not change r.

[latex](2, \frac{π}{3}) = (2, \frac{π}{3} - 2π) = (2, \frac{π}{3} - \frac{6π}{3}) = (2, -\frac{5π}{3})[/latex]

Question 6.3.2: Finding Other Polar Coordinates for a Given Point The point ...