Problem 74 - Precalculus with Limits

Question

In Exercises 65-84, convert the rectangular equation to polar form. Assume \(a>0\). \(3x+5y-2=0\)

Step-by-Step Solution

Answer

The polar form of the rectangular equation \(3x+5y-2=0\) is \(r = \frac{2}{3 \cos θ + 5 \sin θ}\).

Step 1: Identify the relationship between Cartesian coordinates and Polar coordinates

The relationship between Cartesian coordinates (x,y) and Polar coordinates (r,θ) is defined as: \(x = r \cos θ\) and \(y = r \sin θ\)

Step 2: Substitute the relationship into the equation

Substitute \(x = r \cos θ\) and \(y = r \sin θ\) into the equation \(3x+5y-2=0\), we get \(3r \cos θ + 5r \sin θ - 2 =0\)

Step 3: Simplify the equation

Factor out \(r\) to make equation into polar form, we got \((3 \cos θ + 5 \sin θ)r = 2 \). The standard form for polar equations is \(r=f(θ)\). So, the polar form of the given equation is \(r = \frac{2}{3 \cos θ + 5 \sin θ}\).

Key Concepts

Rectangular to Polar Conversion

To convert a rectangular equation into polar form, it's vital to understand the relationship between Cartesian and polar coordinates. Rectangular coordinates are expressed as \((x, y)\), whereas polar coordinates are expressed as \((r, \theta)\). Here, \(x\) and \(y\) are the coordinates on the horizontal and vertical axes, respectively, whereas \(r\) represents the distance from the origin, and \(\theta\) is the angle from the positive x-axis.

The conversion relationships are:

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

Knowing these transformations means we can replace \(x\) and \(y\) with \(r \cos \theta\) and \(r \sin \theta\) in any equation. This substitution is the first step in converting from rectangular to polar form. Once these replacements are made, you can manipulate the equation further to simplify and express it in the standard polar form \(r = f(\theta)\). This conversion facilitates easier analysis, particularly when dealing with curves or circles centered around the origin.

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying expressions, especially in polar equations. These identities are relationships involving trigonometric functions, and they are used to reduce expressions or organize complex terms. Some basic identities are:

• Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta = 1\)
• Double Angle Formulas: \(\cos(2\theta) = \cos^2 \theta - \sin^2 \theta\)
• Sum-to-Product and Product-to-Sum for various angle combinations.

Understanding these identities allows us to rewrite expressions in a more manageable form or solve equations more efficiently. In our exercise, while transforming a rectangular equation into a polar framework, trigonometric identities can simplify intermediary steps and lead to a clearer final expression. They essentially help in combining or breaking down trigonometric terms, laying the groundwork for finding the exact relationships needed in polar coordinates.

Equation Manipulation

Once the rectangular form has been correctly substituted with polar equivalents, the next step is equation manipulation. This involves rearranging and simplifying the equation to suit the standard polar form \(r = f(\theta)\).

Consider our example, after substituting we got: \(3r \cos \theta + 5r \sin \theta - 2 =0\). Notice \(r\) is common in the first two terms. We can factor \(r\) out to simplify the expression:

• \((3 \cos \theta + 5 \sin \theta)r = 2\)

After factoring, isolate \(r\) on one side to yield the standard form of the polar equation:

• \(r = \frac{2}{3 \cos \theta + 5 \sin \theta}\)

This manipulation helps us reveal the underlying relationship between \(r\) and \(\theta\), making it easier to analyze graphically or solve further. In polar form, you're essentially solving for the radius \(r\) in terms of the angle \(\theta\). Thus, mastering equation manipulation enhances your proficiency in transitioning between rectangular and polar analyses.