Problem 35 - Contemporary Precalculus

Question

Find a polar equation that is equivalent to the given rectangular equation. $$x^{2}+y^{2}=25$$

Step-by-Step Solution

Answer

Answer: The equivalent polar equation for the given rectangular equation is \(r^2 = 25\).

Step 1: Write the Rectangular Equation

We are given the rectangular equation as: $$x^{2}+y^{2}=25$$

Step 2: Convert Rectangular to Polar Coordinates

We can replace x and y coordinates with their polar counterparts using the transformation formulas: $$ r^2 \cos^2 \theta + r^2 \sin^2 \theta = 25 $$

Step 3: Simplify the Polar Equation

Now, factor out the r^2 term: $$ r^2 (\cos^2 \theta + \sin^2 \theta) = 25 $$ Since the identity \(\cos^2 \theta + \sin^2 \theta = 1\) holds true, we have: $$ r^2 = 25 $$

Step 4: Final Polar Equation

And finally, our equivalent polar equation is: $$ r^2 = 25 $$

Key Concepts

Rectangular to Polar Conversion

Understanding the process of converting rectangular coordinates to polar coordinates is essential for dealing with equations that describe circles, ellipses, or more complex shapes. Rectangular coordinates, often expressed as (x, y), are common in problems involving Euclidean geometry.
Polar coordinates, on the other hand, are expressed in terms of a radius and angle, usually written as (r, θ). This system is particularly useful for solving problems that involve circular symmetry.

To convert from rectangular to polar:

• Recall that the connection between the systems is defined by the formulas:
  \( x = r \cos \theta \) and \( y = r \sin \theta \).
• For distances, the equation \( r^2 = x^2 + y^2 \) is used, which means that the distance from the origin can be calculated by finding the square root of the sum of the squares of x and y.
• The angle \( \theta \) can be found by \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \), provided \( x eq 0 \).

By mastering these conversions, you can easily transform and handle all sorts of geometric and mathematical problems.

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying and solving equations, especially when working with polar coordinates. These identities allow us to manipulate expressions and make difficult problems much more manageable.
Key identities to remember include:

• **Pythagorean identities:** These include \( \cos^2 \theta + \sin^2 \theta = 1 \). This identity is very useful, as it simplifies expressions by reducing the number of terms involved.
• **Angle sum and difference identities:** Such as \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \) and \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \). These are quite useful in breaking down complex angles.
• **Double angle identities:** These include \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) and \( \sin 2\theta = 2\sin \theta \cos \theta \), easing the process of handling trigonometric functions of angles that are twice as large.

Being comfortable with these and other trigonometric identities is indispensable, as they provide the means to simplify expressions that convert from complex to more familiar forms.

Coordinate Transformation

Coordinate transformation is the application of converting one coordinate system into another. This process is especially useful when one coordinate system simplifies the calculations over another.
For instance, in the given problem, the rectangular equation \( x^2 + y^2 = 25 \) can be cumbersome when dealing with angles and radii directly related to the circle's origin. When converted to polar coordinates,

• The transformation simplifies the expression into \( r^2 = 25 \), elegantly capturing the circular nature of the problem.
• The task reduced here illustrates the transformation from a dispersed system into one that tightly embraces symmetry, easing angles' calculations and radii.

Coordinate transformation fundamentally changes how we perceive problems, harnessing the power of symmetry and geometric features inherent within the concept.

Mathematical Problem-Solving

Mathematical problem-solving involves various strategies, including the use of alternative coordinate systems for clarity and simplification. In this case, converting the rectangular equation of a circle into a polar equation focuses on efficiency and elegance.
To excel at problem-solving, embrace these techniques:

• **Identify appropriate systems:** Know when to switch coordinate systems. Polar coordinates offer powerful simplicity for circular patterns.
• **Use identities and transformations:** Leverage trigonometric identities and coordinate transformations to reduce complexity.
• **Iterate solutions:** Sometimes revisiting and refining your approaches with these tools brings about deeper insights and clarity.

Being adept at recognizing contexts where these problem-solving strategies shine will transform challenges into manageable tasks, demonstrating the elegance and power of mathematical reasoning.