Polar to Cartesian and vice versa equation conversion practice problems

Steps to Solve

1. Understanding Polar and Cartesian Coordinates

Polar coordinates are represented as $(r, \theta)$ where $r$ is the distance from the origin and $\theta$ is the angle from the positive x-axis. Cartesian coordinates are represented as $(x, y)$.

2. Conversion from Polar to Cartesian

To convert from polar to Cartesian coordinates, use the following formulas:

• $x = r \cos(\theta)$
• $y = r \sin(\theta)$

This means that if you have a point in polar coordinates, you can find its Cartesian coordinates using the angle and the radius.

3. Example of Polar to Cartesian Conversion

Let's say we have polar coordinates $(5, \frac{\pi}{3})$.

Using the formulas:

• For $x$: $$ x = 5 \cos\left(\frac{\pi}{3}\right) = 5 \times \frac{1}{2} = 2.5 $$

• For $y$: $$ y = 5 \sin\left(\frac{\pi}{3}\right) = 5 \times \frac{\sqrt{3}}{2} \approx 4.33 $$

Thus, in Cartesian coordinates, it becomes $(2.5, 4.33)$.

4. Conversion from Cartesian to Polar

To convert from Cartesian to polar coordinates, use these formulas:

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

Therefore, given Cartesian coordinates, you can determine the polar coordinates using these equations.

5. Example of Cartesian to Polar Conversion

Suppose we have Cartesian coordinates $(3, 4)$.

Using the conversion formulas:

• For $r$: $$ r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

• For $\theta$: $$ \theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 0.93 , \text{radians} $$

So, in polar coordinates, it becomes $(5, 0.93)$.