To convert the rectangular coordinates $(1, -3)$ into polar coordinates, we use the following formulas:

1. Radius (r): $r = \sqrt{x^2 + y^2}$ Where $x = 1$ and $y = -3$.

2. Angle (θ): $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ This will give the angle in radians. If the point lies in the correct quadrant, no further adjustment is necessary. If it’s in the second or third quadrant, adjustments will be made.

Step 1: Calculate the radius (r)

$r = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \approx 3.16$

Step 2: Calculate the angle (θ)

$\theta = \tan^{-1}\left(\frac{-3}{1}\right) = \tan^{-1}(-3)$

Using a calculator: $\theta \approx -1.249 \text{ radians}$

Since the point $(1, -3)$ lies in the fourth quadrant, the angle $-1.249$ radians is correct, and there is no need to adjust further.

Final Answer:

• Polar Coordinates: $(3.16, -1.25)$, rounded to the nearest hundredth.

Would you like a deeper explanation of the steps or further clarification?

Here are some related questions you might find useful:

1. How do you convert between degrees and radians?
2. What is the significance of the angle's sign in polar coordinates?
3. How do you determine the quadrant in which a point lies?
4. How can you convert polar coordinates back into rectangular coordinates?
5. What would be the polar coordinates if the point was in the second quadrant?

Tip: Polar coordinates are helpful for problems involving circular motion, oscillations, and complex numbers!