Problem 58 - Algebra and Trigonometry

Question

Convert each rectangular equation to a polar equation that expresses r in terms of \(\theta\). $$ x^{2}=6 y $$

Step-by-Step Solution

Answer

The polar equation is \(r = 6 \tan(\theta) \sec(\theta)\) except possibly when \(r = 0\) or \(\theta = k \pi\), where \(k\) is an integer.

Step 1: Identify Rectangular Equation

The rectangular equation provided is \(x^{2}=6 y\). This equation is given in rectangular form, and our goal is to convert it to polar form.

Step 2: Replace x and y With Polar Equations

Replace the \(x\) and \(y\) variables with their respective polar equations. The equations are \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Therefore, replacing in the original equation gives: \((r \cos(\theta))^{2}=6 (r \sin(\theta))\). The left side simplifies to \(r^{2} \cos^{2}(\theta)\), and the right side to \(6r \sin(\theta)\). Therefore, the equation becomes \(r^{2} \cos^{2}(\theta) = 6r \sin(\theta)\).

Step 3: Express r in terms of theta

Next step is to express \(r\) in terms of \(\theta\). Sundividing through by \(r \sin(\theta)\) (but being aware that this wouldn't be valid if \(r = 0\) or \(\theta = k \pi\), where \(k\) is an integer) gives: \(r= \frac{6 \sin(\theta)}{\cos^2(\theta)} = 6 \tan(\theta) \sec(\theta)\). So the required polar equation is \(r = 6 \tan(\theta) \sec(\theta)\), except possibly when \(r = 0\) or \(\theta = k \pi\).

Key Concepts

Rectangular Equation

A rectangular equation is an equation that makes use of the Cartesian coordinate system. In this system, points are defined by the coordinates

• \(x\)
• \(y\)

based on their distances to the vertical and horizontal axis.
This setup is particularly useful for graphing linear, quadratic, or higher-order polynomial equations. Typically, a rectangular equation will look something like
\(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.For example, in the given problem, we start with the rectangular equation \(x^2 = 6y\). This particular style of writing expresses

• \(y\)

in terms of

• \(x\)

, which is commonly encountered in analytic geometry.Understanding rectangular equations and their graph representation can provide valuable insights when transitioning to alternative coordinate systems such as polar coordinates. This is because they allow for a direct method to see how changes in

• \(x\)
• \(y\)

affect the rest of the equation.
Being comfortable with converting these equations into another form, like polar equations, enriches your capacity to solve more complex problems efficiently.

Conversion to Polar Coordinates

The conversion from a rectangular equation to a polar equation involves using relationships between the two coordinate systems. In polar coordinates, a point is determined by

• \(r\)
• \(\theta\)

where

• \(r\) represents the radial distance from the origin
• \(\theta\) represents the angle measured from the positive x-axis.

Here’s how to make the conversion:For conversion, utilize the fundamental polar-to-rectangular relations:

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

By replacing \(x\) and \(y\) with these polar equivalents in an equation, the transformation begins.
In our initial example, we replace \(x^2 = 6y\) with polar terms to arrive at

• \(r^2 \cos^2(\theta) = 6r \sin(\theta)\)

This is a key step where we translate a familiar Cartesian setup into its polar counterpart, which revolves around the distance and direction from a central point of origin.
From here, our task is to manipulate the equation so that \(r\) is isolated and thus expressed solely in terms of angle \(\theta\). This involves carefully simplifying the expression, respecting the balance and symmetry of multiplication and division. Remember, polar conversion may be undefined at particular points, such as when \(r = 0\) or \(\theta = k\pi\), where \(k\) is any integer.

Trigonometric Identities

To solve the polar equation, an understanding of trigonometric identities is also required. Trigonometric identities help by providing relationships between trigonometric functions which simplify complex trigonometric expressions. Here are some key identities that are particularly useful:

• Pythagorean identity: \(\cos^2(\theta) + \sin^2(\theta) = 1\)
• Tangent and secant relations: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) and \(\sec(\theta) = \frac{1}{\cos(\theta)}\)

Notice how the conversion of the equation \(r^2 \cos^2(\theta) = 6r \sin(\theta)\) into the expression \(r = 6 \tan(\theta) \sec(\theta)\) showcases the roles these identities play.By using \(\tan(\theta)\) and \(\sec(\theta)\), it amalgamates two trigonometric ratios into a single expression. This is quite powerful, making it easier to manage and solve trigonometric equations.
Employ these identities strategically, as they often reduce the complexity of many equations and shed light on the underlying relationships between the components of trigonometric expressions.Whether simplifying polynomial forms or translating between coordinate systems, trigonometric identities are staple tools in any mathematician's toolkit. By mastering these identities, navigating the often complex world of mathematics becomes more straightforward and intuitive.