Problem 2 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank. The equation \(r=2 \cos \theta\) represents a ______.

Step-by-Step Solution

Answer

The equation \(r=2 \cos \theta\) represents a circle.

Step 1: Recognize the pattern

The equation given is \(r=2 \cos \theta\). This is a common form of a polar equation, and its kind is generally recognized by the cosine term's presence. The presence of a cosine term like this in a polar equation often suggests that the graph is a circle or a circle-like shape.

Step 2: Recall characteristics of polar curves

Polar equations of the form \(r=a \cos \theta\) or \(r=a \sin \theta\) (where \(a\) is a non-zero constant) represent circles centered on the x-axis or y-axis respectively. Here, the polar equation is \(r=2 \cos \theta\), so it fits in the form of \(r=a \cos \theta\) (a = 2). We can therefore infer that this equation will generate a circle.

Step 3: Verify

To be certain, one could graph the equation or convert it to Cartesian coordinates. In the polar equation \(r=2 \cos \theta\), we could substitute \(r^2 = x^2 + y^2\) and \(y = r \sin \theta\) to see if it matches the standard form of a circle's equation. However, for this task, Step 2 is sufficient.

Key Concepts

Cosine Function

The cosine function is a fundamental aspect of trigonometry, which is the study of the relationships between the sides and angles of triangles. In polar coordinates, the cosine function plays a crucial role in defining the position of points based on their angles. - The function, denoted by \(\cos \theta\), outputs the ratio between the adjacent side and the hypotenuse of a right triangle. - In the unit circle, the cosine of an angle \( \theta \) is equivalent to the \(x\)-coordinate of the corresponding point on the circle.Understanding the cosine function is key because it helps determine the horizontal component of points plotted in polar coordinates, where each point has a distance \(r\) from the origin and an angle \(\theta\) from a reference direction.

Circle in Polar Coordinates

Polar coordinates offer a unique way to describe circles using equations like \(r = a \cos \theta\) and \(r = a \sin \theta\), where \(a\) represents the circle's diameter. These equations are particularly useful when focusing on the geometric aspects of curves related to circular motion.- In an equation such as \(r = 2 \cos \theta\), the presence of \(\cos \theta\) indicates the circle is oriented along the \(x\)-axis.- The parameter \(a\) here is 2, making the equation depict a circle with a diameter of 2, centered at the point \((1, 0)\) in Cartesian coordinates.Recognizing these patterns allows for easy visualization and interpretation of shapes within the polar coordinate system.

Converting to Cartesian Coordinates

Converting polar equations into Cartesian coordinates can often clarify the geometric shape represented. The conversion formulas facilitate this transformation and are given by:

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

To convert \(r = 2 \cos \theta\) into its Cartesian form, follow these steps:1. Use the conversion equation for \(x\), which is \(x = r \cos \theta\). Substituting the polar equation gives \( x = 2 \cos^2 \theta \).2. Recall that \(r^2 = x^2 + y^2\). To remove \(\theta\), use the identity \(\cos^2 \theta = \frac{r^2 - y^2}{r^2} \), and then substitute.Ultimately, this exercise reveals that the polar equation \(r = 2 \cos \theta\) indeed describes a circle in the Cartesian plane with center \((1, 0)\) and radius 1.