Problem 49 - 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 \(49-52 .\) $$\sqrt{2} x+\sqrt{2} y=6$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Cartesian form

The given equation is in Cartesian form: \( \sqrt{2} x + \sqrt{2} y = 6 \).

Step 2: Convert Cartesian coordinates to polar coordinates

Recall the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute these into the equation: \( \sqrt{2}(r \cos \theta) + \sqrt{2}(r \sin \theta) = 6 \).

Step 3: Factor out the common factor

Factor out \( r \) from the expression: \( \sqrt{2}r (\cos \theta + \sin \theta) = 6 \).

Step 4: Simplify and solve for \( r \)

Rearrange to solve for \( r \): \( r (\cos \theta + \sin \theta) = \frac{6}{\sqrt{2}} \). This simplifies to \( r (\cos \theta + \sin \theta) = 3\sqrt{2} \).

Step 5: Express in polar form \( r \cos(\theta - \theta_0) = r_0 \)

Recognize that \( \cos(\theta - \theta_0) = \cos \theta \cos \theta_0 + \sin \theta \sin \theta_0 \). Here, compare: \( \cos \theta + \sin \theta = \sqrt{2}\left(\frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta \right) \).

Step 6: Determine \( \theta_0 \)

Identify that \( \cos \theta_0 = \frac{1}{\sqrt{2}} \) and \( \sin \theta_0 = \frac{1}{\sqrt{2}} \), meaning \( \theta_0 = \frac{\pi}{4} \) (or 45 degrees).

Step 7: Write the polar equation

Finally, express the polar equation as \( r \cos(\theta - \frac{\pi}{4}) = 3\sqrt{2} \).

Key Concepts

Polar Equations

Polar equations are mathematical expressions used to define curves or shapes in a polar coordinate system. In a polar system, each point on a plane is determined by a distance from a central point, called the pole, and an angle from a fixed direction, usually the positive x-axis (or 0 degrees). The general form of a polar equation can significantly vary based on the symmetry and type of curve, such as circles, spirals, or rose curves.

Key aspects of polar equations include:

• The variable "r" represents the radial distance to the point from the pole.
• The angle "\(\theta\)" represents the direction from the pole, measured in radians or degrees.
• Equations can be expressed as functions of "r" in terms of "\(\theta\)" or "\(\theta\)" in terms of "r".

In some cases, polar equations can resemble linear equations in Cartesian coordinates, as seen in expressions like \( r \cos(\theta - \theta_0) = r_0 \). This specific form is useful for describing lines in a polar context, highlighting the clear link between radial distance, angle rotation, and Cartesian equivalents.

Cartesian to Polar Conversion

The conversion from Cartesian coordinates to polar coordinates bridges two different methods of defining points in a plane. Cartesian coordinates represent points using "x" and "y" values, which correspond to horizontal and vertical distances from the origin. In contrast, polar coordinates use "r" and "\(\theta\)", the radial distance and angle from the pole.

To convert a Cartesian point \((x, y)\) into polar form, the following formulas apply:

• Radial distance: \( r = \sqrt{x^2 + y^2} \)
• Angle: \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \)

Using these conversions, you can translate equations or points from one system to another. For lines given in Cartesian form, substitution of \(x = r \cos \theta\) and \(y = r \sin \theta\) allows for a transformation into polar form. This methodology was applied in the solution procedure, facilitating the expression of the given line in a polar context.

Trigonometric Identities

Trigonometric identities play a crucial role in solving polar equations and converting between coordinate systems. These mathematical formulas establish relationships between trigonometric functions, aiding in the simplification and transformation of expressions.

Some key trigonometric identities useful in this context include:

• Sum of angles: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• Difference of angles: \( \cos(\theta - \theta_0) = \cos \theta \cos \theta_0 + \sin \theta \sin \theta_0 \)
• Pythagorean Identity: \( \cos^2 \theta + \sin^2 \theta = 1 \)

In the provided exercise, the identity for the cosine of the difference of angles is primarily utilized to connect a sum of trigonometric functions to a single cosine expression. This facilitates the conversion of a multi-term expression into the specific form needed for polar lines. Recognizing these identities' utility allows students to maneuver complex equations into more manageable forms, streamlining their problem-solving process.