Problem 4 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

Plot the point 1 unit to the right of the origin on the x-axis.

Step 1: Understanding Polar Coordinates

Polar coordinates are represented in the form \(r, \theta\). Here, \(r\) is the distance from the origin (pole), and \(\theta\) is the angle with the positive x-axis, measured in radians. For the given point \( (1,0) \), \(r = 1\) and \(\theta = 0\).

Step 2: Locating the Radius

Since \(r = 1\), the point is located at a distance of 1 unit from the origin. We'll need to remember this when moving from the pole (origin) in the direction given by the angle.

Step 3: Identifying the Angle

The angle \(\theta = 0\) indicates that the point lies directly on the positive x-axis. Since there is no angle change, it stays on the x-axis starting from the origin.

Step 4: Moving Along the Axis

With an angle of \(0\) radians, move from the origin horizontally to the right along the x-axis to a distance of 1 unit.

Step 5: Plotting the Point

Finally, plot the point at \( (1, 0) \) on the positive x-axis. This is the same as the Cartesian coordinates \( (1, 0) \) in this special case, since the point lies on the x-axis.

Key Concepts

Plotting Points in Polar Coordinates

In polar coordinates, plotting points involves understanding a different system from the Cartesian coordinate system. Instead of using x and y coordinates, polar coordinates use a combination of a radius and an angle. The radius, denoted as \(r\), describes the distance from the pole, which is analogous to the origin in Cartesian coordinates. The angle, \(\theta\), describes how far to rotate from the positive x-axis, measured in the counterclockwise direction.

• The point \((1,0)\) is plotted by first identifying \(r = 1\), indicating how far it is from the origin.
• The angle \(\theta = 0\) indicates the direction along the axis, which is directly to the right.

The plotting process involves drawing a line from the origin at the specified angle and marking the point at the determined distance \(r\). In this case, the point is simply one unit away along the x-axis.
The polar coordinate system is particularly useful in cases involving symmetry, circular motion, or where rotations are a primary focus.

Distance from Origin in Polar Coordinates

The concept of distance in polar coordinates is straightforward. The distance \(r\) from the origin tells you exactly how far your point is from the pole, which serves as the central reference point. It's always a non-negative number and represents the radius of the circle on which the point lies.

• For example, with \(r = 1\) in the point \((1,0)\), the point is precisely 1 unit away from the origin.

This distance plays a crucial role in determining the position of any point expressed in polar coordinates. Whether your radius is large or small will dictate how far from the origin your point is situated, effectively mapping circles in this coordinate system. Interestingly, if \(r = 0\), your point converges exactly at the origin, regardless of the angle \(\theta\).

Angle Measurement in Radians

Understanding radians is essential for accurately determining angles in polar coordinates. Unlike degrees, radians provide a more straightforward measure of angles, especially in the context of mathematical calculations.

• A full circle is \(2\pi\) radians, while a half circle is \(\pi\) radians.
• An angle of \(0\) radians, as in our point \((1,0)\), aligns with the positive x-axis.

It's crucial to note that radians provide a direct link between the distance around a circle and the radius, making it easier to work with equations involving rotation or periodic functions. Moreover, when expressing angles in radians, calculations become smoother because of the natural connection to the arc length and radius of circles.