Problem 43 - Algebra and Trigonometry

Question

Convert the equation to polar form. $$ y=x^{2} $$

Step-by-Step Solution

Answer

The polar form is \( r = \tan(\theta) \sec(\theta) \).

Step 1: Understand the Cartesian Equation

The given equation is in Cartesian coordinates: \( y = x^2 \). In this equation, \( y \) is expressed in terms of \( x \). To convert it to polar form, we need to express both \( x \) and \( y \) in terms of polar coordinates \( r \) and \( \theta \).

Step 2: Substitute Polar Coordinates

The polar coordinate system is related to Cartesian coordinates by the relationships \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). Substitute these into the Cartesian equation to begin the conversion process: \( r \sin(\theta) = (r \cos(\theta))^2 \).

Step 3: Simplify the Equation

Simplify the equation from Step 2: \( r \sin(\theta) = r^2 \cos^2(\theta) \). Divide both sides by \( r \) (assuming \( r eq 0 \)) to isolate \( r \): \( \sin(\theta) = r \cos^2(\theta) \).

Step 4: Express in Polar Form

Rearrange the equation from Step 3 to express \( r \) in terms of \( \theta \): \( r = \frac{\sin(\theta)}{\cos^2(\theta)} \), which can also be written as \( r = \tan(\theta) \sec(\theta) \). This is the polar form of the given equation.

Key Concepts

Cartesian Coordinates

Cartesian coordinates are used to determine each point uniquely in a plane with a pair of numerical values. The values represent the distance of the point along two perpendicular axes: usually called the x-axis (horizontal) and y-axis (vertical). For example, any point in the Cartesian plane can be specified as \((x, y)\).

In the context of equations like \(y = x^2\), these values describe the parabola's shape and position in the 2D plane.

• The x-value represents the horizontal distance from the origin.
• The y-value shows how far up or down the point is relative to the x-axis.

Using Cartesian coordinates is very helpful in algebra and geometry, as they provide a straightforward way to illustrate relationships and positions.Understanding this system is crucial when it comes to converting to other coordinate systems like polar coordinates.

Equation Conversion

Equation conversion from Cartesian to polar coordinates involves substituting the Cartesian values with their polar equivalents. This step is essential when working on problems that require a different representation of an equation, such as when analysis in polar form simplifies the understanding or solving of the problem.

The basic relationships between Cartesian and polar coordinates are:

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

To convert an equation from Cartesian to polar form, substitute these relationships into the equation.For example, starting from \(y = x^2\), substitute to get \(r \sin(\theta) = (r \cos(\theta))^2\).Once substituted, the second step is to simplify the equation.By dividing through by \(r\) (provided \(r eq 0\)), you further isolate terms and make further conversions easier. This process allows you to express the equation cleanly in terms of polars \(r\) and \(\theta\).

Trigonometric Identities

Trigonometric identities play a vital role when converting equations into polar form. These identities help simplify complex expressions and facilitate the conversion process.

Some of the key trigonometric identities useful in this context include:

• \(\sin(\theta)\) and \(\cos(\theta)\) as natural substitutes from the polar conversions.
• The identity \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) simplifies the expression of some converted equations.
• The secant identity \(\sec(\theta) = \frac{1}{\cos(\theta)}\) aids in rewriting certain expressions.

Using these identities, for example, you can rewrite \(r = \frac{\sin(\theta)}{\cos^2(\theta)}\) as \(r = \tan(\theta) \sec(\theta)\).Understanding these identities provides insight into how trigonometric functions relate to each other, ensuring proper conversion and simplification during equation transformations.