Question

Find two other pairs of polar coordinates of the given polar coordinate, one with r > 0 and one with r < 0.

Then plot the point.

(a) (5, 7π/4)

(r, θ) ( ) (r > 0)
(r, θ) ( ) (r < 0)

(b) (−6, π/2)

(r, θ) ( ) (r > 0)
(r, θ) ( ) (r < 0)

(c) (5, −2)

(r, θ) ( ) (r > 0)
(r, θ) ( ) (r < 0)

Answer 1

Answer:

A) (5, 7π/4):   (5, 15π/4) and (-5, 3π/4)

B) (−6, π/2):   (−6, 5π/2) and (6, -π/2)

C) (5, −2):       (5, 2π-2) and (-5, -π-2)

Step-by-step explanation:

To find pair of coordinates for (r>0), add 2π to corresponding θ. For (r<0) subtract π from given angle to find second pair of coordinate.

A)  (5, 7π/4)

For (r>0)

θ =  7π/4 + 2π

θ =  15π/4

Point is (5, 15π/4)

For (r<0)

θ =  7π/4 - π

θ =  3π/4

Point is (-5, 3π/4)

As it can be seen in Fig 1

(5, 7π/4)=(5, 15π/4)=(-5, 3π/4)

B)  (−6, π/2)

For (r>0)

θ =  π/2 + 2π

θ =  5π/2

Point is (6, 5π/2)

For (r<0)

θ =  π/2 - π

θ =  -π/2

Point is (-6, -π/2)

As shown in fig 2

(-6, π/2) = (6, 5π/2) = (-6, -π/2)

C) (5, −2)

For (r>0)

θ =  -2 + 2π

Point is (5, 2π-2)

For (r<0)

θ =  -2 - π

Point is (-5, -π)

As shown in Fig. 3

(5, −2) = (5, 2π-2) = (-5, -π)

Answer 2

The origin of a polar coordinate is referred to as a pole

The other pairs of a polar coordinate are:

• $\mathbf{(5, \frac{11}{4}\pi) \ and\ (-5, \frac{3}4\pi) }$.
• $\mathbf{(6, \frac{3}{2}\pi) \ and\ (-6, -\frac{\pi}2) }$.
• $\mathbf{(5, -2 + \pi) \ and\ (5, -2 - \pi) }$

The other pair of a polar coordinate is calculated as:

$\mathbf{(r,\theta) \to (r, \theta \pm n\pi)}$

Where:

$\mathbf{n \ne 0}$

To do this, we make use of n = 1.

Note that, n can take any value.

$\mathbf{(a)\ (5, \frac{7}{4}\pi)}$

When r > 0

$\mathbf{(5, \frac{7}{4}\pi) \to (5, \frac{7}{4}\pi + \pi)}$

$\mathbf{(5, \frac{7}{4}\pi) \to (5, \frac{11}4\pi)}$

When r < 0

$\mathbf{(-5, \frac{7}{4}\pi) \to (-5, \frac{7}{4}\pi - \pi)}$

$\mathbf{(-5, \frac{7}{4}\pi) \to (-5, \frac{3}4\pi)}$

So, the other pair of polar coordinates are:

$\mathbf{(5, \frac{11}{4}\pi) \ and\ (-5, \frac{3}4\pi) }$

$\mathbf{(b)\ (-6, \frac{\pi}{2})}$

When r > 0

$\mathbf{(6, \frac{\pi}{2}) \to (6, \frac{\pi}{2} + \pi)}$

$\mathbf{(6, \frac{\pi}{2}) \to (6, \frac{3}{2}\pi)}$

When r < 0

$\mathbf{(-6, \frac{\pi}{2}) \to (-6, \frac{\pi}{2} - \pi)}$

$\mathbf{(-6, \frac{\pi}{2}) \to (-6, -\frac{\pi}{2})}$

So, the other pair of polar coordinates are:

$\mathbf{(6, \frac{3}{2}\pi) \ and\ (-6, -\frac{\pi}2) }$

$\mathbf{(c)\ (5, -2)}$

When r > 0

$\mathbf{(5,-2) \to (5, -2 + \pi)}$

When r < 0

$\mathbf{(5,-2) \to (5, -2 - \pi)}$

So, the other pair of polar coordinates are:

$\mathbf{(5, -2 + \pi) \ and\ (5, -2 - \pi) }$

Read more about polar coordinates at:

https://brainly.com/question/23679942