Question

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (-0.8,-2.1) $$

Short answer

The polar coordinates are approximately \(2.25, 4.345\).

Step-by-step solution

- Understand the Conversion Formula

To convert from rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), use the formulas \[r = \sqrt{x^2 + y^2}\] and \[\theta = \tan^{-1} \left( \frac{y}{x} \right)\].

- Calculate the Radial Distance

Calculate the radial distance \(r\) using the formula \[r = \sqrt{x^2 + y^2}.\] With \(x = -0.8\) and \(y = -2.1\), we get \[r = \sqrt{(-0.8)^2 + (-2.1)^2} = \sqrt{0.64 + 4.41} = \sqrt{5.05} \approx 2.25.\]

- Calculate the Angle

Compute the angle \(\theta\) using \[\theta = \tan^{-1} \left( \frac{y}{x} \right).\] With \(x = -0.8\) and \(y = -2.1\), we find \[\theta = \tan^{-1} \left( \frac{-2.1}{-0.8} \right) = \tan^{-1} (2.625).\] This yields an angle in the 1st quadrant. However, since both \(x\) and \(y\) are negative, the point is actually in the 3rd quadrant. Therefore, add \pi\ to the angle obtained to find the correct angle in the 3rd quadrant: \[\theta \approx \tan^{-1}(2.625) + \pi \approx 1.203 + 3.142 \approx 4.345.\]

- State the Polar Coordinates

The polar coordinates \(r, \theta\) are approximately \(2.25, 4.345.\)

Key concepts

polar coordinates

Polar coordinates are an alternative way of locating points on a plane. Instead of using the traditional Cartesian or rectangular coordinates \((x, y)\), polar coordinates use a distance from a reference point (often called the origin) and an angle from a reference direction.
This system is particularly useful in scenarios where the relationship between points is more naturally described by angles and distances, such as in circular motion or when using radar. This means that any point can be described by two values: the radial distance \(r\) and the angle \(\theta\).

• The radial distance \(r\) tells us how far a point is from the origin.
• The angle \(\theta\) indicates the direction of the point relative to the positive x-axis.

When converting from rectangular to polar coordinates, we use trigonometric functions to calculate these two values based on the given \((x, y)\) coordinates.

radial distance

The radial distance, often represented by \(r\), is the straightforward part of the conversion process. It measures the direct distance from the origin to the point in question. This distance is always a positive value or zero.
The formula to calculate the radial distance is: \[ r = \sqrt{x^2 + y^2} \] Here’s how it works:

• Square both the x-coordinate and the y-coordinate.
• Add these squared values together.
• Take the square root of the result to find \(r\).

For example, for the point \((-0.8, -2.1)\), we calculate: \[ r = \sqrt{(-0.8)^2 + (-2.1)^2} = \sqrt{0.64 + 4.41} = \sqrt{5.05} \approx 2.25 \] Thus, the radial distance \(r\) is approximately 2.25.

angle calculation

Calculating the angle \(\theta\) involves a bit more attention to detail, especially considering which quadrant the point lies in. Initially, you use the following formula to find the angle in radians: \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \] However, this only gives you an angle in the correct range if the point is in the first quadrant.
To get the correct angle for all quadrants, you need to make adjustments based on the signs of the x and y coordinates:

• If both x and y are positive, the point lies in the first quadrant, and the angle calculated is already correct.
• If x is negative and y is positive, the point is in the second quadrant. Add \(\pi\) to the calculated angle.
• If both x and y are negative, the point is in the third quadrant. Again, add \(\pi\) to the calculated angle.
• If x is positive and y is negative, the point is in the fourth quadrant. Subtract the calculated angle from \(2\pi\).

Using our example \((-0.8, -2.1)\), we initially find: \[ \theta = \tan^{-1} \left( \frac{-2.1}{-0.8} \right) = \tan^{-1}(2.625) \].
Since both coordinates are negative, the point is in the third quadrant. We then adjust by adding \(\pi\): \[ \theta \approx 1.203 + 3.142 \approx 4.345 \] Thus, the angle \(\theta\) is approximately 4.345 radians.