Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-6 x=0$$

Step-by-Step Solution

Answer

The polar form of the given equation, assuming \(a<0\), is \(r=6\cos(\theta + \pi)\).

Step 1: Identify and express variables in polar form

You're given the equation \(x^{2}+y^{2}-6 x=0\). In polar coordinates, \(x\) and \(y\) are expressed as \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). So, substitute these into the given equation.

Step 2: Substitute and Simplify

The equation becomes \(r^2 \cos^2(\theta) + r^2 \sin^2(\theta) - 6r\cos(\theta)=0\). Regrouping terms, we have \(r^2[\cos^2(\theta) + \sin^2(\theta)] - 6r\cos(\theta)=0\). Recall that the identity \(\cos^2(\theta) + \sin^2(\theta) = 1\), so, the equation simplifies to \(r^2 - 6r\cos(\theta)=0\).

Step 3: Finalize the conversion

In order to completely convert the equation to polar form, the equation can be factored to \(r(r - 6\cos(\theta))=0\). So, the polar forms of the equation are \(r=0\) and \(r=6\cos(\theta)\). However, because we're assuming \(a<0\), we need to flip the orientation of \(\cos(\theta)\) to match \(a\), so our final equation is \(r=6\cos(\theta + \pi)\).

Key Concepts

Rectangular to Polar Conversion

Converting a rectangular equation to polar form involves switching from the Cartesian coordinate system, where points are defined by \((x, y)\), to the polar coordinate system, where points are represented by \((r, \theta)\). This can make certain types of problems, especially those involving circles or periodic shapes, easier to solve. For conversion:

• The variable \(x\) is expressed in terms of polar coordinates as \(x = r \cos(\theta)\).
• The variable \(y\) changes to \(y = r \sin(\theta)\).

Substitute these expressions into the given equation, transforming the equation from an \((x, y)\) format to an \((r, \theta)\) format. With this exercise, after substitution, collect terms intelligently to get an equation in \(r\) and \(\theta\) only.Break down the equation step by step, ensuring you maintain the relationships between \(x\), \(y\), \(r\), and \(\theta\) accurately.

Trigonometric Identities

Trigonometric identities are essential tools in converting between coordinate systems as they simplify the expressions in equations. Some of the most fundamental identities include:

• \(\sin^2(\theta) + \cos^2(\theta) = 1\), known as the Pythagorean Identity.
• Basic definitions like \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).

In our exercise, we used the identity \(\cos^2(\theta) + \sin^2(\theta) = 1\) to simplify the expression \(r^2[\cos^2(\theta) + \sin^2(\theta)]\) to \(r^2\). This crucial step reduces complexity and makes it easier to manipulate the equation into its desired polar form.Understanding these identities aids in recognizing patterns and simplifies the algebraic manipulation needed to solve the problem.

Algebraic Manipulation

Algebraic manipulation involves the strategic rearranging and simplifying of mathematical expressions. It's a crucial skill when converting between rectangular and polar forms, as it helps to isolate variables and factor equations correctly.In this conversion exercise, the equation has been rearranged and simplified through several steps:

• After substitution, terms are grouped by \(r^2\) and \(-6r\cos(\theta)\).
• Applying trigonometric identities helps clear terms, simplifying \(r^2 - 6r\cos(\theta) = 0\).
• Factoring leads to \(r(r - 6\cos(\theta))=0\), which provides potential solutions: \(r=0\) and \(r=6\cos(\theta)\).

Recognizing when and how to manipulate equations is essential for both solving and understanding mathematical problems in polar conversions. This mastery aids in achieving clear, concise results suitable for interpreting the behavior of equations in both polar and rectangular contexts.