Problem 68 - Precalculus Mathematics for Calculus

Question

Convert the polar equation to rectangular coordinates. $$\cos 2 \theta=1$$

Step-by-Step Solution

Answer

The rectangular equation is the x-axis, where \( y = 0 \).

Step 1: Understand the Polar Equation

The given polar equation is \( \cos 2\theta = 1 \). Our goal is to convert this equation into rectangular coordinates (\(x, y\)).

Step 2: Use Double Angle Identity

Remember the trigonometric identity \( \cos^2\theta - \sin^2\theta = \cos 2\theta\). Here, \( \cos 2\theta = 1 \) implies \( \cos^2\theta - \sin^2\theta = 1 \).

Step 3: Relationship with Rectangular Coordinates

Recall the relationships between polar and rectangular coordinates: \( x = r\cos\theta \) and \( y = r\sin\theta \). Also, note that \( r^2 = x^2 + y^2 \).

Step 4: Replace Trigonometric Functions

Using the relationships, \(\cos\theta = \frac{x}{r}\) and \(\sin\theta = \frac{y}{r}\). Thus, \( \cos^2\theta = \frac{x^2}{r^2} \) and \( \sin^2\theta = \frac{y^2}{r^2} \).

Step 5: Substitute and Simplify

Substitute these into the identity: \( \frac{x^2}{r^2} - \frac{y^2}{r^2} = 1 \). Since \( r^2 = x^2 + y^2 \), the equation becomes \( \frac{x^2}{x^2 + y^2} - \frac{y^2}{x^2 + y^2} = 1 \).

Step 6: Simplify the Fraction

Combine the fractions: \( \frac{x^2 - y^2}{x^2 + y^2} = 1 \). Cross-multiply to solve for the equation: \( x^2 - y^2 = x^2 + y^2 \). This simplifies to \( -y^2 = y^2 \).

Step 7: Solve the Resulting Equation

This further simplifies to \( 0 = 2y^2 \), which implies \( y^2 = 0 \). Thus, \( y = 0 \), so the rectangular equation simplifies to the x-axis.

Key Concepts

Polar Equations

Polar equations make use of a different coordinate system compared to the more familiar rectangular or Cartesian coordinates. In this system, each point on the plane is determined by a distance from a reference point and an angle from a reference direction. These are represented as \( (r, \theta) \), where \( r \) is the radius (the distance from the pole, equivalent to the origin in rectangular coordinates) and \( \theta \) is the angle.

Polar coordinates are especially useful for problems involving circular or rotational symmetry. They simplify the representation of circles and spirals. In contrast, rectangular coordinates, expressed as \( (x, y) \), are better for linear problems.

Converting from polar to rectangular coordinates involves using the relationships \( x = r\cos\theta \) and \( y = r\sin\theta \). These equations allow one to translate polar expressions into the rectangular plane, making it easier to visualize certain problems without having to leave the rectangular grid behind.

Trigonometric Identities

Trigonometric identities play a crucial role in converting polar equations into rectangular coordinates. They are equations involving trigonometric functions that are true for all values of involved variables. For this exercise, the double angle identity \( \cos 2\theta = \cos^2\theta - \sin^2\theta \) was particularly important.

This identity allowed us to express \( \cos 2\theta \) in terms of \( \cos\theta \) and \( \sin\theta \), laying the groundwork for converting the equation into a form involving \( x \) and \( y \).

• The double angle identity is useful when simplifying equations or expressions involving trigonometric functions.
• It can help in integrating or differentiating trigonometric equations.
• Investigating properties of trigonometric functions often requires the use of these identities.

Leveraging these identities effectively helps in understanding the geometric interpretations and further simplifies complex problems, making polar equations easier to manage.

Coordinate Transformation

The process of coordinate transformation involves changing the way a point or equation is represented from one coordinate system to another. In this scenario, we moved from polar coordinates to rectangular coordinates. Understanding transformations is crucial in mathematics, as different systems offer various advantages depending on the problem at hand.

Transforming the equation \( \cos 2\theta = 1 \) required knowledge of how to interchange variables using established relationships:

• \( x = r\cos\theta \)
• \( y = r\sin\theta \)
• \( r^2 = x^2 + y^2 \) which combines the above two relations.

By expressing \( \cos\theta \) and \( \sin\theta \) in terms of \( x \) and \( y \), the original equation was translated into the rectangular equation.

Coordinate transformations not only allow for easier computation but also provide insights such as simplifying integral calculus problems, examining symmetries, and solving differential equations across various applications in physics and engineering.