Question

$$\text {Use the results of Exercises } 74-75 \text { to}$$ $$\text { describe and graph the following circles.}$$ $$r^{2}-4 r \cos (\theta-\pi / 3)=12$$

Short answer

Based on the above solution, the given polar equation represents a circle with its center at (1, √3) in Cartesian coordinates and a radius of 4. To graph the circle, plot the center point (1, √3) and draw a circle with radius 4 centered at this point.

Step-by-step solution

Recall standard circle equation in polar coordinates

The standard form of a circle equation in polar coordinates is \((r-R\cos(\theta-\phi))^2+(R\sin(\theta-\phi))^2=a^2\), where \((R,\phi)\) is the polar representation of the center of the circle, and \(a\) is its radius. Notice that only the \(x\)-part (cosine part) of the equation is given in our exercise.

Identify the equation's center and radius

Compare the given equation \(r^2 - 4r\cos(\theta - \pi/3) = 12\) to the standard form. We can rewrite it as \((r-2\cos(\theta-\pi/3))^2=16\). Thus, the circle has a center at \((R, \phi) = (2, \pi/3)\) and radius \(a = 4\).

Convert the center to Cartesian coordinates

The center of the circle is given in polar coordinates as \((2, \pi/3)\). To graph the circle, we can convert the center into Cartesian coordinates using the conversion formulas \(x = r\cos{\theta}\) and \(y = r\sin{\theta}\). So, the Cartesian coordinates of the center are \((x, y) = (2\cos(\pi/3), 2\sin(\pi/3)) = (1, \sqrt{3})\).

Graph the circle

Now that we have the center \((1, \sqrt{3})\) and radius \(4\), we can graph the circle. Here are the main steps for this graph: 1. Plot the Cartesian center \((1, \sqrt{3})\). 2. Draw the circle with the given radius \(4\) centered at \((1, \sqrt{3})\). Now, you have correctly described and graphed the circle represented by the polar equation \(r^2 - 4r\cos(\theta - \pi/3) = 12\).

Key concepts

Circle Equation

In mathematics, the equation of a circle is a fundamental concept, particularly when dealing with geometry and trigonometry. The circle equation can be represented in different forms depending on the coordinate system used. When working in polar coordinates, the general form of a circle equation is expressed as:\[(r - R\cos(\theta - \phi))^2 + (R\sin(\theta - \phi))^2 = a^2\]Here, \((R, \phi)\) denotes the polar coordinates of the circle's center, while \(a\) is the radius of the circle. This formula intuitively represents the distance between any point \((r, \theta)\) on the circle's circumference and the center point. The left side of the equation essentially measures this distance, while the right side represents the constant radius squared.
To identify the center and radius from a polar circle equation, you compare it with this standard form, extracting values for the center and the radius as needed.In our specific example, the polar circle equation is \(r^2 - 4r\cos(\theta - \pi/3) = 12\). Comparing this with the standard form reveals that the circle has a center at \((2, \pi/3)\) and a radius of \(4\). This shows that polar circle equations effectively translate the circle properties into a format that's usable in trigonometry and geometry problems.

Cartesian Coordinates

Cartesian coordinates are an essential concept in graphing and solving mathematical equations, particularly those involving shapes like circles. They allow us to describe the position of points in a plane using a pair of numbers: \((x, y)\). These coordinates are based on two perpendicular axes known as the x-axis and y-axis. In many contexts, it is beneficial to convert from polar coordinates to Cartesian coordinates to facilitate graphing or analysis.To convert polar coordinates \((r, \theta)\) into Cartesian coordinates, you use the following formulas:

• \(x = r \cos{\theta}\)
• \(y = r \sin{\theta}\)

This transformation is highly valuable when a circle's equation is initially given in polar form, as it allows you to plot the circle in a standard Cartesian plane.For our exercise, converting the center of the circle \((2, \pi/3)\) from polar to Cartesian form yields \((x, y) = (1, \sqrt{3})\). These coordinates will serve as the center point in Cartesian terms for graphing the circle on a typical x-y coordinate grid.

Graphing Circles

Graphing circles, whether in Cartesian or polar coordinates, involves visualizing a round, symmetrical shape defined by a center and a radius. In the Cartesian coordinate system, this is generally easier because the circle's position can be directly plotted using easily understandable \((x, y)\) coordinates.To graph a circle given its center \((x_0, y_0)\) and radius \(r\), follow these steps:

• Locate the circle's center on the Cartesian plane by pinpointing the coordinates \((x_0, y_0)\).
• From this center, measure a distance equal to the radius in all directions to outline the circle's boundary.
• Draw a smooth curve connecting these points to form the complete circle.

For the exercise under analysis, the circle is centered at \((1, \sqrt{3})\) with a radius of \(4\). Placing the center involves marking the coordinate \((1, \sqrt{3})\) on the plane, approximately at \(1\) unit along the x-axis and \(\sqrt{3}\), or about \(1.732\) units along the y-axis.
From this calculated center, you then draw the circle by ensuring every point on the circumference is exactly \(4\) units from the center, forming a perfect circle.