Chapter H.1, Problem 1E

(a)

To determine

To plot: the point and find two other pairs of polar coordinate of $r>0$ and $r<0$ .

Answer to Problem 1E

Therefore, other coordinates are $\left[\left(2,\frac{7\pi }{3}\right)\right]$ and $\left[\left(-2,\frac{4\pi }{3}\right)\right]$ .

Explanation of Solution

Given:

Polar coordinates are: $(2,\pi /3)$

Calculation:

Consider the provided point $\left(2,\frac{\pi }{3}\right)$

It is plotted in the polar coordinates as follows:

  

To find other pairs of polar coordinates of this point, either add multiples of $2\pi$ (a full circle) to the angle, or add odd multiples of $\pi$ (a half circle) to the angle and reverse the sign on the radius.

Therefore, other coordinates are $\left[\left(2,\frac{7\pi }{3}\right)\right]$ and $\left[\left(-2,\frac{4\pi }{3}\right)\right]$ .

Conclusion:

Therefore, other coordinates are $\left[\left(2,\frac{7\pi }{3}\right)\right]$ and $\left[\left(-2,\frac{4\pi }{3}\right)\right]$ .

(b)

To determine

To plot: the point and find two other pairs of polar coordinate of $r>0$ and $r<0$ .

Answer to Problem 1E

Hence, the two other coordinates are $\left[\left(1,\frac{5\pi }{4}\right)\right]$ and $\left[\left(-1,\frac{\pi }{4}\right)\right]$ .

Explanation of Solution

Given:

Polar coordinates are: $(1,-3\pi /4)$

Calculation:

Consider the provided point $\left(1,\frac{-3\pi }{4}\right)$

It is plotted in the polar coordinates as follows:

  

To find other pairs of polar coordinates of this point, either add multiples of $2\pi$ (a full circle) tothe angle, or add odd multiples of $\pi$ (a half circle) to the angle and reverse the sign on the radius.

Hence, the two other coordinates are $\left[\left(1,\frac{5\pi }{4}\right)\right]$ and $\left[\left(-1,\frac{\pi }{4}\right)\right]$ .

Conclusion:

Hence, the two other coordinates are $\left[\left(1,\frac{5\pi }{4}\right)\right]$ and $\left[\left(-1,\frac{\pi }{4}\right)\right]$ .

(c)

To determine

To plot: the point and find two other pairs of polar coordinate of $r>0$ and $r<0$ .

Answer to Problem 1E

Therefore, other coordinates are $\left[\left(-1,\frac{5\pi }{2}\right)\right]$ and $\left[1,\frac{3\pi }{2}\right)$ .

Explanation of Solution

Given:

Polar coordinates are: $(-1,\pi /2)$

Calculation:

Consider the provided point $\left(-1,\frac{\pi }{2}\right)$

It is plotted in the polar coordinates as follows:

  

To find other pairs of polar coordinates of this point, either add multiples of $2\pi$ (a full circle) to the angle, or add odd multiples of $\pi$ (a half circle) to the angle and reverse the sign on the radius.

Therefore, other coordinates are $\left[\left(-1,\frac{5\pi }{2}\right)\right]$ and $\left[1,\frac{3\pi }{2}\right)$

Conclusion:

Therefore, other coordinates are $\left[\left(-1,\frac{5\pi }{2}\right)\right]$ and $\left[1,\frac{3\pi }{2}\right)$ .