Problem 3 - Algebra and Trigonometry

Question

Plot the point that has the given polar coordinates. $$ (4, \pi / 4) $$

Step-by-Step Solution

Answer

Plot a point 4 units out at a 45° angle from the positive x-axis.

Step 1: Identify Polar Coordinates

The polar coordinates given are \((r, \theta) = (4, \pi/4)\), where \(r\) is the radius and \(\theta\) is the angle in radians.

Step 2: Convert Angle to Cartesian Plane

The angle \(\pi/4\) radians is equivalent to 45 degrees. In the polar coordinate system, this angle is measured counterclockwise from the positive x-axis. So, point will be positioned at 45 degrees in the Cartesian plane but at a distance \(r = 4\) from the origin.

Step 3: Draw the Radius

From the origin, draw a line segment that is 4 units long at an angle of 45 degrees (\(\pi/4\) radians) to the positive x-axis. This puts the point in the first quadrant.

Step 4: Plot the Point

The endpoint of the 4-unit line segment is the location of the point with polar coordinates (4, \(\pi/4\)). Plot this point by marking the end of the line on graph or coordinate plane, in the first quadrant.

Key Concepts

Cartesian Plane

The Cartesian Plane is a two-dimensional plane defined by a horizontal axis, often referred to as the x-axis, and a vertical axis, known as the y-axis. These two axes intersect at a point called the origin. Points on this plane are determined by ordered pairs \(x, y\), indicating their positions along these axes.
When dealing with polar coordinates, like \(r, \theta\), it's common to convert them to Cartesian coordinates for ease of plotting on the Cartesian Plane. The conversion is achieved using the formulas:

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

This conversion provides the x and y coordinates, enabling us to plot the point on the Cartesian Plane effortlessly.
Understanding how to navigate between these two systems is essential for accurately representing points and functions in different contexts.

Radians

Radians are a unit of angular measurement used in many areas of mathematics, particularly in trigonometry and geometry. Unlike degrees, where a full circle is divided into 360 parts, radians offer a more natural unit by using \(2\pi\) to represent a complete circle.
The formula for converting degrees to radians is \( \text{radians} = \dfrac{\text{degrees} \times \pi}{180} \). Conversely, to convert from radians to degrees, use \( \text{degrees} = \dfrac{\text{radians} \times 180}{\pi} \).
In the context of polar coordinates, angles are typically given in radians. For instance, \( \pi/4 \) radians is equivalent to 45 degrees, and this angle is key to plotting points in both polar and Cartesian representations. Understanding radians helps in recognizing angles' real-world and mathematical significance as they naturally connect arc lengths and radii in circles.

First Quadrant

The First Quadrant is one of the four sections divided by the x and y axes in the Cartesian Plane.
It is located at the top-right corner where both x and y values are positive. When plotting polar coordinates, it's important to recognize in which quadrant the point lies.
With \(\theta = \pi/4\), the angle of 45 degrees signifies that the point is indeed in the First Quadrant of the Cartesian Plane. Recognizing this helps prevent errors in plotting and ensures you accurately find the position in the graph.
This quadrant represents forward and upward movement, aligning well with radii led by positive angles from the x-axis, integral in understanding where points will situate themselves on a plane.