Question

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (1,-1) $$

Short answer

The polar coordinates are \( \left( \sqrt{2}, -\frac{\pi}{4} \right) \).

Step-by-step solution

Understand the relationship between Rectangular and Polar coordinates

Rectangular coordinates \((x, y)\) can be converted to polar coordinates \((r, \theta)\), where \(r\) is the radius and \(\theta\) is the angle. The formulas are: \[ r = \sqrt{x^2 + y^2} \]\[ \theta = \arctan\left( \frac{y}{x} \right) \]

Calculate the radius \(r\)

Substitute the given coordinates \((1, -1)\) into the formula:\[ r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

Calculate the angle \(\theta\)

Substitute the values into the angle formula:\[ \theta = \arctan\left( \frac{-1}{1} \right) = \arctan(-1) \]The angle whose tangent is -1 is \(-\frac{\pi}{4}\). Hence, \( \theta = -\frac{\pi}{4} \).

Determine the polar coordinates

Combine the calculated radius and angle to form the polar coordinates: \((r, \theta) = \left( \sqrt{2}, -\frac{\pi}{4} \right)\).

Key concepts

Rectangular Coordinates

Rectangular coordinates describe the position of a point in a plane using two values: the x-coordinate and the y-coordinate. Imagine a point on a graph. The point's horizontal location is determined by the x-coordinate, while its vertical location is determined by the y-coordinate.
For example, in the coordinate pair \(1, -1\), the x-coordinate is 1 and the y-coordinate is -1.
These values represent how far the point is from the origin (0,0), which is the center of the graph, along the x-axis and y-axis, respectively.
To visualize this, draw a line 1 unit to the right from the origin (for the x-value) and then another line 1 unit downward from there (for the y-value). You will find the point \(1, -1\).

Radius Calculation

To convert rectangular coordinates to polar coordinates, we need to calculate the radius, denoted as \(r\). The radius is the distance from the origin to the given point.
To find the radius, we use the formula: \[ r = \sqrt{x^2 + y^2} \] Here, \(x\) and \(y\) are the given rectangular coordinates.
Plugging in our values \(x = 1\) and \(y = -1\): \[ r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \] Therefore, the radius (\(r\)) is \(\sqrt{2}\).
The radius helps us understand how far away the point is from the origin, regardless of its direction.

Angle Calculation

Next, we need the angle \(\theta\) to complete the polar coordinates. The angle is measured counterclockwise from the positive x-axis to the line connecting the origin to our point.
The formula to calculate the angle is: \[ \theta = \arctan\left( \frac{y}{x} \right) \] Using our rectangular coordinates \(1\) and \(-1\), we get: \[ \theta = \arctan\left( \frac{-1}{1} \right) = \arctan(-1) \]
The arctangent of \(-1\) is \(-\frac{\pi}{4}\). So, \( \theta = -\frac{\pi}{4} \)
The angle tells us the direction of the point from the origin in radians. It's important to understand the unit circle and how angles relate to positions on this circle.

Coordinate Conversion

Finally, we combine our calculated radius and angle to convert rectangular coordinates to polar coordinates. Polar coordinates are represented as \((r, \theta)\).
For the point \(1, -1\), we found: \[ r = \sqrt{2} \] and \[ \theta = -\frac{\pi}{4} \] Therefore, the polar coordinates are: \[ \left( \sqrt{2}, -\frac{\pi}{4} \right) \] This shows us the point in a different way: using its distance from the origin and its angle from the x-axis.
Converting coordinates helps in various fields, including engineering and physics, to simplify complex problems or provide different perspectives on data.