Question

Which polar coordinates represent the same point as the rectangular coordinate left parenthesis 1 comma negative 3 right parenthesis? Select all that apply.
(1 point)
Responses

left parenthesis start root 10 end root comma 71.6 degrees right parenthesis
Image with alt text: left parenthesis start root 10 end root comma 71.6 degrees right parenthesis

left parenthesis start root 10 end root comma 288.4 degrees right parenthesis
Image with alt text: left parenthesis start root 10 end root comma 288.4 degrees right parenthesis

left parenthesis negative start root 10 end root comma 116.6 degrees right parenthesis
Image with alt text: left parenthesis negative start root 10 end root comma 116.6 degrees right parenthesis

left parenthesis negative start root 10 end root comma 108.4 degrees right parenthesis

Answer 1

To convert the rectangular coordinates \((1, -3)\) into polar coordinates, we first calculate the radius \(r\) and the angle \(\theta\).

1. **Calculate the radius \(r\)**:
\[
r = \sqrt{x^2 + y^2} = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

2. **Calculate the angle \(\theta\)**:
The angle can be found using the arctangent function:
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-3}{1}\right) = \tan^{-1}(-3)
\]
This angle is in the fourth quadrant since \(x > 0\) and \(y < 0\).

\(\tan^{-1}(-3)\) gives an angle that is approximately \(-71.6\) degrees. To express the angle in standard position (between \(0\) and \(360\) degrees), we can add \(360\) degrees:
\[
\theta \approx 360 - 71.6 = 288.4 \text{ degrees}
\]

Now we have the polar coordinates:
- One representation is \((\sqrt{10}, 288.4^\circ)\).

Other equivalent polar coordinates can be found by adding or subtracting \(360\) degrees or reflecting the point. In this case, since the angle can also be expressed positively, we can reflect through the origin to find other equivalent angles:
- \((-\sqrt{10}, 116.6^\circ)\) corresponds to reflecting along the origin which is \(180 + 288.4 = 468.4 - 360 = 108.4\).

So the polar coordinates that represent the same point \((1, -3)\) are:
- \((\sqrt{10}, 288.4^\circ)\)
- \((-\sqrt{10}, 116.6^\circ)\)
- \((-\sqrt{10}, 108.4^\circ)\)

### Final Responses:
1. \((\sqrt{10}, 288.4^\circ)\)
2. \((-\sqrt{10}, 116.6^\circ)\)
3. \((-\sqrt{10}, 108.4^\circ)\)

**Available selections:**
- \((\sqrt{10}, 288.4^\circ)\)
- \((-\sqrt{10}, 116.6^\circ)\)
- \((-\sqrt{10}, 108.4^\circ)\)

The correct selections are:
- \((\sqrt{10}, 288.4^\circ)\)
- \((-\sqrt{10}, 116.6^\circ)\)
- \((-\sqrt{10}, 108.4^\circ)\)