Problem 14 - Precalculus

Question

Convert to rectangular form. $$r^{2}=1 /\left(3+\cos ^{2} \theta\right)$$

Step-by-Step Solution

Answer

Rectangular form: \((x^2 + y^2)^2 = 4x^2 + 3y^2\).

Step 1: Understanding the Polar Equation

The given equation is in polar form: \( r^2 = \frac{1}{3 + \cos^2 \theta} \). Our goal is to convert this into rectangular form, which involves expressing the equation using \( x \) and \( y \) coordinates instead of \( r \) and \( \theta \).

Step 2: Use Relationships Between Polar and Rectangular Coordinates

We use the relationships: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). Let's first replace \( r^2 \) with \( x^2 + y^2 \) in the initial equation, leading to \( x^2 + y^2 = \frac{1}{3 + \cos^2 \theta} \).

Step 3: Express \( \cos^2 \theta \) in Rectangular Terms

Recall that \( \cos \theta = \frac{x}{r} \), so \( \cos^2 \theta = \left(\frac{x}{r}\right)^2 = \frac{x^2}{r^2} \). Substitute \( r^2 = x^2 + y^2 \), giving \( \cos^2 \theta = \frac{x^2}{x^2 + y^2} \).

Step 4: Substitute Back into the Main Equation

Substitute \( \cos^2 \theta = \frac{x^2}{x^2 + y^2} \) into the main equation: \( x^2 + y^2 = \frac{1}{3 + \frac{x^2}{x^2 + y^2}} \).

Step 5: Simplify the Expression

Find a common denominator for \( 3 + \frac{x^2}{x^2 + y^2} \): \( 3(x^2 + y^2) + x^2 \). This gives \( x^2 + y^2 = \frac{1}{\frac{3x^2 + 3y^2 + x^2}{x^2 + y^2}} \). Simplify this equation to find an expression solely in terms of \( x \) and \( y \).

Step 6: Clear the Fraction and Rearrange

Multiply both sides by the denominator: \((x^2 + y^2)^2 = x^2 + 3(y^2 + x^2)\). Simplify to get \((x^2 + y^2)^2 = 4x^2 + 3y^2\) as the final rectangular form.

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in the plane using a radius and an angle. Instead of using two perpendicular distances like in the Cartesian system, polar coordinates use:

• Radius ( ): This is the distance from the origin to the point.
• Angle ( heta): This is the angle from the positive x-axis to the line connecting the point to the origin.

Polar coordinates are particularly useful when dealing with circular or spiral shapes, where the angle and radius can describe the position more naturally.

An important part of converting between polar and rectangular coordinates is understanding the relationships between them:

• To find the x-coordinate: \( x = r \cos \theta \)
• To find the y-coordinate: \( y = r \sin \theta \)
• The relationship between \( r \) and \( x, y \) is: \( r^2 = x^2 + y^2 \)

These relationships allow us to translate polar equations, like the one provided, into rectangular equations by replacing \( r \) and \( \theta \) with \( x \) and \( y \). This translation can often make further analysis or graphing easier.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points using two numbers

• x: The horizontal distance from the origin.
• y: The vertical distance from the origin.

These coordinates form the standard method for plotting points in a two-dimensional plane.

The conversion from polar to rectangular coordinates involves using the relationships between polar radius and angle, and the rectangular x and y. Once these relationships are applied, the original polar equation, often written in terms of \( r \) and \( \theta \), becomes an equation in terms of \( x \) and \( y \). This conversion step was shown in the exercise.

In our example, the original polar equation \( r^2 = \frac{1}{3 + \cos^2 \theta} \) was converted to the rectangular equation \( (x^2 + y^2)^2 = 4x^2 + 3y^2 \). Such conversions are critical when working across different coordinate systems, allowing us to apply different methods and analyses to solve problems.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the trigonometric functions to one another. They are very useful when converting equations between polar and rectangular forms.

For converting polar coordinates, an important identity is the cosine identity:

• \( \cos^2 \theta = \left(\frac{x}{r}\right)^2 \)

This is derived by knowing that \( \cos \theta = \frac{x}{r} \). Once this identity is applied, it allows us to transform parts of the polar equation into terms of \( x \) and \( y \).

Using trigonometric identities helps simplify the expression of the polar equation as it is transformed into a rectangular equation. In the step-by-step example provided, using \( \cos \theta = \frac{x}{r} \) led us to convert \( \cos^2 \theta \) into a fraction involving \( x \), \( y \), and \( r^2 \). This identity allows the equation to be rewritten and resultantly simplified into a completely rectangular form, as demonstrated in the solution. These identities are essential tools in mathematical conversions and problem solving.