Problem 25 - The Calculus with Analytic Geometry

Question

Find a cartesian equation of the graph having the given polar equation.\(r^{2}=\cos \theta\)

Step-by-Step Solution

Answer

\[ x (x^{2} + y^{2})^{\frac{3}{2}} = x \] or simpler form: \[ x^{2} + y^{2} = 1\]

Step 1: Recall the polar to Cartesian transformation

To transform from polar to Cartesian coordinates, recall the equations: \[ x = r \cos \theta \] \[ y = r \sin \theta \] \[ r^{2} = x^{2} + y^{2} \]

Step 2: Substitute the given polar equation

Given the equation \[ r^{2} = \cos \theta \] we can substitute the polar to Cartesian transformations.

Step 3: Express \( \cos \theta \) in Cartesian form

Since \( \cos \theta = \frac{x}{r} \), we can replace it in the equation: \[ r^{2} = \frac{x}{r} \]

Step 4: Simplify the equation

Multiply both sides by \( r \) to eliminate the denominator: \[ r^{3} = x \]

Step 5: Substitute \( r^{2} \)

We know \( r^{2} = x^{2} + y^{2} \). So, \[ r = \sqrt{x^{2} + y^{2}} \] Substitute this value back into the equation: \[ (\sqrt{x^{2} + y^{2}})^{3} = x \]

Step 6: Simplify to the final Cartesian equation

Apply cube to the equation: \[ x (x^{2} + y^{2})^{\frac{3}{2}} = x \]. Simplify this to get the Cartesian form: \[ x (x^{2} + y^{2}) = x^{3} \]

Step 7: Final touch

Since both sides have x, if \( x eq 0 \), divide both sides by \( x \): \[ x^{2} + y^{2} = 1\]. Otherwise, we acknowledge the case when \( x = 0 \)

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance and an angle. Instead of using x and y like in the Cartesian system, we use two values: \( r \) for the radius (distance from the origin) and \( \theta \) for the angle (measured from the positive x-axis).
If you think of a circle, \( r \) tells you how far from the center you are, and \( \theta \) tells you the direction.

To locate any point:

• Start at the origin.
• Move \( r \) units away from the origin in the direction specified by the angle \( \theta \).

Now you know your location in the plane!

Cartesian Coordinates

Cartesian coordinates use two values, x and y, to specify a point in a plane. Imagine graph paper: x is how far right or left you go, and y is how far up or down you go.

In simple terms:

• x tells you where you are horizontally.
• y tells you where you are vertically.

This is great for straight lines or rectangular shapes, but for curves or angles, polar coordinates can be easier.

Coordinate Transformation

Transforming from polar to Cartesian coordinates means converting \( r \) and \( \theta \) into x and y.
The equations you'll need are:

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

By using these equations, you can rewrite any polar equation in the Cartesian format.

For instance:

• Given the polar equation \( r^{2} = \cos \theta \)
• First, express \( \cos \theta = \frac{x}{r} \)
• Then, substitute \( r^{2} = x^{2} + y^{2} \) into the polar equation.
• After simplifying, you get the Cartesian form of the equation.

r^2 = x^2 + y^2

One of the most important equations when converting between polar and Cartesian coordinates is \( r^{2} = x^{2} + y^{2} \).
This comes from the Pythagorean theorem and represents the relationship between the radius (r) and the x and y coordinates.

For instance:

• Given \( r^{2} = \cos \theta \), you substitute \( r^{2} \) with \( x^{2} + y^{2} \).
• This simplifies the process of converting polar equations to Cartesian form.

It is the backbone of understanding coordinate transformations and gets you to the correct Cartesian equation.