To find the polar coordinates for the point $(-3, 4)$ in rectangular form, we need to convert it to the form $(r, \theta)$, where:

• $r$ is the radial distance (distance from the origin),
• $\theta$ is the angle (measured counterclockwise from the positive x-axis).

Step 1: Calculate the radial distance $r$

The radial distance $r$ is given by the formula: $r = \sqrt{x^2 + y^2}$ where $x = -3$ and $y = 4$.

Substitute the values of $x$ and $y$: $r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ So, $r = 5$.

Step 2: Calculate the angle $\theta$

The angle $\theta$ can be found using the inverse tangent function, considering the quadrant the point lies in.

The angle is calculated by: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ For $x = -3$ and $y = 4$, the ratio $\frac{y}{x} = \frac{4}{-3} = -\frac{4}{3}$.

Using a calculator, we find: $\theta = \tan^{-1}\left(-\frac{4}{3}\right) \approx -53.13^\circ$

However, the point $(-3, 4)$ lies in the second quadrant, where the angle is positive. To adjust for this, we add $180^\circ$ to the calculated angle: $\theta = -53.13^\circ + 180^\circ = 126.87^\circ$

Thus, the polar coordinates for the point $(-3, 4)$ are approximately: $(r, \theta) = (5, 126.87^\circ)$

Final Answer:

The polar coordinates of the point $(-3, 4)$ are $(5, 126.87^\circ)$.

Would you like further explanation on this? Feel free to ask if something is unclear!

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Related questions:

1. How do you convert polar coordinates back to rectangular coordinates?
2. How do you determine the quadrant in which a point lies based on its rectangular coordinates?
3. What is the general form for expressing polar coordinates?
4. How do you calculate the angle in polar coordinates when the point lies on the x-axis or y-axis?
5. How does the sign of $r$ affect the representation of polar coordinates?

Tip: Remember that angles in polar coordinates are usually expressed in degrees or radians. Ensure you are consistent with the unit used for $\theta$.