Problem 4 - Algebra and Trigonometry

Question

Plot the point that has the given polar coordinates. $$ (1,0) $$

Step-by-Step Solution

Answer

The point (1,0) in polar coordinates is plotted at (1, 0) on the Cartesian plane.

Step 1: Understand Polar Coordinates

Polar coordinates are defined as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis (measured in radians). In this problem, we have the polar coordinates \((1, 0)\).

Step 2: Convert Polar to Cartesian Coordinates

To plot a point given polar coordinates \((r, \theta)\), convert it to Cartesian coordinates \((x, y)\) using the formulas:\[ x = r \cdot \cos(\theta) \]\[ y = r \cdot \sin(\theta) \]Substituting the given values, \(r = 1\) and \(\theta = 0\), we find the Cartesian coordinates:\[ x = 1 \cdot \cos(0) = 1 \]\[ y = 1 \cdot \sin(0) = 0 \]Thus, the Cartesian coordinates of the point are \((1, 0)\).

Step 3: Plot the Point on a Cartesian Plane

Now that we have the Cartesian coordinates \((1, 0)\), we can plot the point on a Cartesian plane. The \(x\)-coordinate is 1, and the \(y\)-coordinate is 0, meaning the point lies on the positive x-axis, exactly 1 unit from the origin.

Key Concepts

Cartesian coordinates

Imagine the world laid out on a grid, where each point is identified by two numbers, known as Cartesian coordinates. These coordinates are typically expressed as \(x, y\), representing the horizontal and vertical positions respectively. In the conversion from polar to Cartesian coordinates, the aim is to translate from a circular path to this straight-line grid.

For example, when given polar coordinates like \((r, \theta) = (1, 0)\), we can find the corresponding Cartesian coordinates using:

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

In our case, the calculations yield the point \(1, 0\), securely placing it one unit away from the origin along the x-axis.

This concept is beneficial for translating circular motion or positional data into easily understandable forms, like graph plots that we encounter daily.

radial distance

Radial distance is the distance from the center of a circle or origin point in a polar coordinate system. It is denoted by \(r\).

In polar coordinates, the point \((r, \theta)\) means you start at the origin and move straight outwards a distance of \(r\) units. This distance is similar to the radius of a circle. When \(r = 1\), as in our exercise problem, it signifies that the point is exactly one unit away from the center.

This measure is crucial for understanding how far away a point is, without focusing on which directions it's oriented towards. Understanding radial distance allows you to comprehend how polar coordinates function as a system based on distance and angle.

angle measurement

Angle measurement in polar coordinates is represented by \(\theta\), indicating the direction from the positive x-axis where the point lies. Angles are essential for calculating direction in a circular manner.

In the given coordinate \(1, 0\), \(\theta = 0\) radians implies the point is aligned directly on the positive x-axis. The angle tells you which direction from the center to start counting the radial distance, essentially describing a position on an imaginary circle.

• Angles in polar coordinates can be measured in radians or degrees; however, radians are usual in mathematics due to their natural fit in calculations involving circles.
• Remember that \(2\pi\) radians complete a full circle.

In effect, understanding angles alongside radial distances leads to pinpointing locations precisely in a polar coordinate system.