Problem 9 - Thomas Calculus

Question

Find a polar equation in the form \(r \cos \left(\theta-\theta_{0}\right)=r_{0}\) for each of the lines in Exercises \(9-12\) . $$ \sqrt{2} x+\sqrt{2} y=6 $$

Step-by-Step Solution

Answer

The polar equation is \( r \cos \left(\theta - \frac{\pi}{4}\right) = \frac{3}{\sqrt{2}} \).

Step 1: Identify the Line Equation

Given the line equation: \( \sqrt{2}x + \sqrt{2}y = 6 \). This is a standard line equation in Cartesian form.

Step 2: Convert to Polar Coordinates

Recall that the conversion from Cartesian coordinates to polar coordinates involves using the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute these into the given equation:\[ \sqrt{2}(r \cos \theta) + \sqrt{2}(r \sin \theta) = 6 \].

Step 3: Factor Out r

Factor \( r \) out of the equation: \[ r (\sqrt{2} \cos \theta + \sqrt{2} \sin \theta) = 6 \].

Step 4: Simplify Using Trigonometric Identities

Notice that \( \sqrt{2} \cos \theta + \sqrt{2} \sin \theta \) can be rewritten using a trigonometric formula: \[ \sqrt{2}( \cos \theta + \sin \theta ) = \sqrt{2} \sqrt{2} \cos \left(\theta - \frac{\pi}{4}\right) \]. Therefore, \[ r \cdot 2 \cos \left(\theta - \frac{\pi}{4}\right) = 6 \].

Step 5: Rearrange to Polar Equation Format

Re-arrange the equation into the desired polar form: \[ r \cos \left(\theta - \frac{\pi}{4}\right) = \frac{3}{\sqrt{2}} \], since \( r_0 \) is \( \frac{6}{2} \). This gives us the polar equation.

Key Concepts

Line Equations

A line equation in Cartesian coordinates typically follows the form \( ax + by = c \). In our original problem, the equation \( \sqrt{2}x + \sqrt{2}y = 6 \) represents a straight line. Understanding how to manipulate and interpret line equations is crucial, as they define relationships between the \(x\) and \(y\) values. This specific line equation features coefficients of \( \sqrt{2} \), implying a symmetrical form around both axes due to the equal influence of \(x\) and \(y\) values, making it balanced. When working with line equations, consider the following points:

• Identify coefficients: They directly affect the slope and orientation.
• Right-hand value: Determines how far the line is positioned from the origin.
• Form transformation: From Cartesian to polar as needed for specific applications.

Through correct manipulation, such equations can be reshaped to reveal more about the spatial relationships they define.

Trigonometric Identities

Trigonometric identities are a key tool in simplifying expressions involving angles. These identities allow us to relate linear Cartesian equations to polar equivalents using simple geometric transformations. In the given problem, the trigonometric identity used is the transformation: \[\cos \theta + \sin \theta = \sqrt{2}\cos\left(\theta - \frac{\pi}{4}\right)\]This specific identity combines both sine and cosine terms into a singular cosine term with a shifted angle. It simplifies recognizing patterns between Cartesian and polar forms. It helps reduce the complexity of multi-term trigonometric expressions by breaking them down into simpler components. Other trigonometric identities include:

• Pythagorean Identities: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Angle Sum and Difference: \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \)
• Double Angle: \( \sin(2\theta) = 2\sin\theta\cos\theta \)

Knowing when and how to apply these can significantly streamline problem-solving.

Polar Equation Conversion

Converting an equation from Cartesian to polar coordinates helps express lines and curves in a system defined by radius and angle, rather than horizontal and vertical distances. In this process, we use the transformations \( x = r \cos \theta \) and \( y = r \sin \theta \). When applied to our problem, these conversions take the line \( \sqrt{2}x + \sqrt{2}y = 6 \) and transform it to a polar equation by substituting and factoring the common \(r\). The conversion process includes:

• Substitute \( x \) and \( y \) with their polar equivalents.
• Factor out \( r \): This highlights how the line relates to polar distances.
• Simplify using trigonometric identities: Aligns to the standard polar form.

The ultimate goal is an equation using the format \( r \cos(\theta - \theta_0) = r_0 \). This standardization enables solutions to be plotted in a polar coordinate system and gives insights into the circular symmetry and radial distance characteristics of the line in question.