Problem 50 - Contemporary Precalculus

Question

Find a rectangular equation that is equivalent to the given polar equation. $$r=3 \cos \theta$$

Step-by-Step Solution

Answer

Question: Convert the polar equation r = 3cosθ to a rectangular equation and identify the geometric shape it represents. Answer: The rectangular equation equivalent to the given polar equation is x² = x². This equation represents the entire x-axis in the rectangular coordinate system, which means the geometric shape it represents is a line.

Step 1: Write down the given equation

The given polar equation is: $$r = 3 \cos \theta$$

Step 2: Replace r with x and y

We know that \(x = r \cos \theta\), so we can rewrite the given equation as: $$r = \frac{x}{\cos \theta}$$ Now, replace r in the equation: $$\frac{x}{\cos \theta} = 3 \cos \theta$$

Step 3: Eliminate θ

Now we have an equation with x and θ, but we need to get rid of θ. To do this, we'll use the relation \(r^2 = x^2 + y^2\). Since \(r = \frac{x}{\cos \theta}\), we can write: $$\left(\frac{x}{\cos \theta}\right)^2 = x^2 + y^2$$ Now we can square both sides and simplify: $$\frac{x^2}{\cos^2 \theta} = x^2 + y^2$$ Recall the trigonometric identity \(\cos^2 \theta + \sin^2 \theta = 1\) , We can rewrite \(\cos^2 \theta\) as \(1 - \sin^2 \theta\) : $$\frac{x^2}{1 - \sin^2 \theta} = x^2 + y^2$$

Step 4: Substitute sin θ with y and r

We know that \(y = r \sin \theta\). So, \(\sin \theta = \frac{y}{r}\). Now, substitute \(\sin \theta\) with \(\frac{y}{r}\) in the above equation: $$\frac{x^2}{1 - \left(\frac{y}{r}\right)^2} = x^2 + y^2$$ Since we know that \(r = \frac{x}{\cos \theta}\), substitute r with \(\frac{x}{\cos \theta}\): $$\frac{x^2}{1 - \left(\frac{y}{\frac{x}{\cos \theta}}\right)^2} = x^2 + y^2$$

Step 5: Simplify the equation and eliminate θ

Multiply both sides of the equation by \((1-\sin^2 \theta)\) to eliminate the denominator: $$x^2 = (x^2 + y^2)(1-\sin^2 \theta)$$ Now, substitute back \(\cos^2 \theta\) for \((1-\sin^2 \theta)\) : $$x^2 = (x^2 + y^2) \cos^2 \theta$$ Recalling that \(x = r \cos \theta\), we can solve for \(\cos^2 \theta\): $$\cos^2 \theta = \frac{x^2}{(x^2 + y^2)}$$ Now substitute this expression back in the equation: $$x^2 = (x^2 + y^2) \frac{x^2}{(x^2 + y^2)}$$

Step 6: Simplify the final equation

Cancel out \((x^2 + y^2)\) from both sides: $$x^2 = x^2$$ Finally, we have a rectangular equation equivalent to the given polar equation: $$x^2 = x^2$$ This equation represents the entire x-axis in the rectangular coordinate system. The given polar equation represents all points on the x-axis when converted to rectangular coordinates.

Key Concepts

Polar Coordinate System

The polar coordinate system is an alternative to the traditional rectangular (or Cartesian) coordinate system in which each point on a plane is defined by a distance from a reference point and an angle from a reference direction. This reference point is called the pole (similar to the origin in the Cartesian system), and the reference direction is usually the positive x-axis, known as the polar axis.

In polar coordinates, a point is represented by \(r, \theta\), where \(r\) is the radial distance from the pole, and \(\theta\) is the angular component, measured in radians or degrees from the polar axis. For positive \(r\), \(\theta\) is typically in the range \(0 \le \theta < 2\pi\) radians (or 0 to 360 degrees). This coordinate system is particularly useful in scenarios where symmetry with respect to the pole or circular patterns are present, making it a valuable tool for solving problems in various fields such as physics, engineering, and trigonometry.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. These identities are fundamental to the study of trigonometry and play a key role in a wide range of applications from geometry to calculus. One of the most crucial identities is the Pythagorean identity, which states that for any angle \(\theta\):

\[\cos^2 \theta + \sin^2 \theta = 1\]
This identity allows us to express one trigonometric function entirely in terms of another, facilitating the process of converting between polar and rectangular coordinates. Knowing these identities, and how to manipulate them, is essential for understanding the relationships between trigonometric functions and for simplifying complex expressions.

Converting Polar to Rectangular Coordinates

Converting polar to rectangular coordinates involves using the relationships between the polar and Cartesian systems to express \(r, \theta\) in terms of \(x, y\). The conversion relies on the trigonometric functions sine and cosine, as well as the Pythagorean Theorem.

The x-coordinate is found by multiplying the radial distance by the cosine of the angle: \[x = r \cos \theta\]
The y-coordinate is obtained by multiplying the radial distance by the sine of the angle: \[y = r \sin \theta\]

These formulas are derived from the unit circle and the definition of the trigonometric functions. By using these conversion formulas, we can translate any polar equation or coordinate into its rectangular equivalent, allowing for easier analysis and graphing on the Cartesian plane.

Rectangular Coordinate System

The rectangular (or Cartesian) coordinate system is the most familiar coordinate system where points are located using two perpendicular axes intersecting at a fixed point called the origin. These axes are usually labeled as the x (horizontal) and y (vertical) axes.

Every point in the plane is associated with a pair of numbers \(x, y\), which indicate the positions along the x-axis and y-axis, respectively. The rectangular coordinate system is especially useful for graphing equations and functions, and plotting data in a straightforward, intuitive manner. It provides a framework for calculus, linear algebra, and numerous other fields in mathematics and science. Understanding how to move between the rectangular system and other systems like the polar system is a critical skill in mathematics and related disciplines.