Question

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (0,2) $$

Short answer

(2, \frac{\pi}{2}\)

Step-by-step solution

- Understand the Relationship Between Coordinate Systems

Rectangular coordinates \(x, y\) can be converted to polar coordinates \((r, \theta)\). The polar coordinates are given by the formulas: \( r = \sqrt{x^2 + y^2} \ and \ \theta = \arctan(\frac{y}{x}) \).

- Calculate the Radius \(r\)

Substitute \( x = 0 \) and \( y = 2 \) into the formula for \( r \): \ r = \sqrt{0^2 + 2^2} = \sqrt{4} = 2 \.

- Calculate the Angle \( \theta \)

Substitute \( x = 0 \) and \( y = 2 \) into the formula for \( \theta \): \ \theta = \arctan(\frac{2}{0}) \. Since \ x = 0 \ and \ y = 2 \, the point lies on the positive y-axis, thus \ \theta = \frac{\pi}{2} \.

- Write Down the Polar Coordinates

Combine the calculated values of \( r \) and \( \theta \) to write the polar coordinates. Thus, the polar coordinates are \( (2, \frac{\pi}{2}) \).

Key concepts

Rectangular Coordinates

Rectangular coordinates, represented as \(x, y\), are based on the Cartesian coordinate system. In this system, the position of a point is determined by two values: the horizontal distance from the origin (x) and the vertical distance from the origin (y). For example, the point \(0, 2\) indicates that the point is 0 units along the x-axis and 2 units up along the y-axis.
Understanding how to locate a point using rectangular coordinates is essential before converting to polar coordinates. This system uses perpendicular lines forming a grid, making it intuitive for plotting points, performing algebraic operations, and working with geometric shapes.

Radius Calculation

To convert rectangular coordinates to polar coordinates, we first calculate the radius \(r\). The radius represents the distance of the point from the origin (0, 0) in the coordinate system. The formula to find the radius is: \ r = \sqrt{x^2 + y^2} \
In this example, given the point \(0, 2\), we substitute \(x = 0\) and \(y = 2\) into the formula:
\[ r = \sqrt{0^2 + 2^2} = \sqrt{4} = 2 \]
So, the radius \(r\) is 2. This step is crucial because the radius tells us how far the point is from the origin in the polar coordinate system.

Angle Calculation

After calculating the radius, the next step is to find the angle \(\theta\), which gives the direction of the point relative to the positive x-axis. The angle can be found using the arctangent function: \ \theta = \arctan(\frac{y}{x}) \
Substituting the values from our point, \(0, 2\), we get:
\[ \theta = \arctan(\frac{2}{0}) \]
Since \(x = 0\) and \(y = 2\), the point lies directly on the positive y-axis. By convention, the angle for points on the positive y-axis is \( \frac{\pi}{2} \) radians. Thus,
\[ \theta = \frac{\pi}{2} \]
Understanding how to determine the angle helps us specify the direction part of polar coordinates.

Coordinate Systems Conversion

Conversion between rectangular and polar coordinates involves re-expressing a point from one system into the other. Rectangular coordinates are given by \(x, y\), while polar coordinates use \(r, \theta\) where \(r\) is the radius and \(\theta\) is the angle.
For a given point \(0, 2\):

• Calculate the radius: \ r = 2\
• Calculate the angle: \ \theta = \frac{\pi}{2} \

The polar coordinates are \( (2, \frac{\pi}{2}) \).
This conversion is useful in fields like physics and engineering where circular motion or systems with radial symmetry are analyzed. It allows easier computation of distances and angles, making problem-solving more intuitive in certain contexts.