To find the polar coordinates of the point with rectangular coordinates $(x, y) = (1.9, -2.6)$, we use the following formulas:

1. Radius (r):
   $r = \sqrt{x^2 + y^2}$
2. Angle ($\theta$):
   $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Step 1: Calculate $r$

$r = \sqrt{(1.9)^2 + (-2.6)^2}$ $r = \sqrt{3.61 + 6.76} = \sqrt{10.37} \approx 3.22$

Step 2: Calculate $\theta$

$\theta = \tan^{-1}\left(\frac{-2.6}{1.9}\right)$ $\theta = \tan^{-1}(-1.368)$ Using a calculator: $\theta \approx -53.13^\circ$ Convert this to radians (since polar coordinates are often expressed in radians): $\theta = -53.13^\circ \times \frac{\pi}{180} \approx -0.93 \, \text{radians}.$

Final Answer:

The polar coordinates are approximately: $(3.22, -0.93)$

Would you like further clarification or additional examples? Here are five related questions to explore:

1. How do you convert polar coordinates back to rectangular coordinates?
2. What happens if the angle $\theta$ is expressed in degrees instead of radians?
3. Can you explain why polar coordinates have multiple valid representations?
4. How do you determine the quadrant of the point using $\theta$?
5. How do you handle $\tan^{-1}$ calculations when both coordinates are negative?

Tip: Always check if your angle needs adjustment based on the quadrant of the rectangular coordinates.