Problem 49 - Precalculus with Calculus Previews

Question

Find a polar equation that has the same graph as the given rectangular equation. $$ x^{2}+y^{2}=5 y $$

Step-by-Step Solution

Answer

The polar equation is \(r = 5\sin\theta\).

Step 1: Identify the Rectangular Equation

The given rectangular equation is \(x^2 + y^2 = 5y\). This equation is in terms of \(x\) and \(y\), which are Cartesian coordinates.

Step 2: Express in Polar Coordinates

Identify the relationships between rectangular and polar coordinates: \(x = r\cos\theta\), \(y = r\sin\theta\), and \(x^2 + y^2 = r^2\). Substitute these into the equation: \(r^2 = 5r\sin\theta\).

Step 3: Simplify the Polar Equation

To simplify, factor out an \(r\) on the right: \(r^2 = 5r\sin\theta\) becomes \(r(r - 5\sin\theta) = 0\).

Step 4: Solve for 'r'

Set each factor equal to zero. For \(r = 0\), this is the origin, which doesn't describe the entire graph. For \(r - 5\sin\theta = 0\), solve to find \(r = 5\sin\theta\).

Step 5: Finalize the Polar Equation

Thus, the polar equation equivalent to the given rectangular equation is \(r = 5\sin\theta\). This describes the same graph in polar form.

Key Concepts

Rectangular to Polar Conversion

Converting between rectangular and polar equations is essential in mathematics, especially in coordinate geometry. If you need to transform a rectangular equation (Cartesian coordinates) to a polar equation, knowing the relationship between these two systems is critical. The primary connections include:

• For the rectangular coordinates:
  • \( x = r\cos\theta \)
  • \( y = r\sin\theta \)
  • \( x^2 + y^2 = r^2 \)

To convert a rectangular equation to polar form, simply substitute these expressions. For example, given the rectangular equation \( x^2 + y^2 = 5y \), replace \( x^2 + y^2 \) with \( r^2 \) and \( y \) with \( r\sin\theta \). This substitution transforms the equation into \( r^2 = 5r\sin\theta \). By understanding these expressions, it becomes easier to navigate between these two coordinate systems and solve problems effectively.

Polar Coordinates

Polar coordinates provide an alternative to the typical Cartesian coordinate system. Instead of using an \((x, y)\) pair, polar coordinates use a combination of a distance from the origin \(r\) and an angle \(\theta\) (in radians) from the positive x-axis. This system is particularly useful when dealing with problems involving circular or rotational symmetry.
Some key advantages of using polar coordinates include:

• Simplifying equations for circles and spirals.
• Facilitating calculations involving rotational motion.
• Aiding in understanding the position of points in a two-dimensional plane based on angle and distance.

To visualize a point in polar coordinates, imagine moving \(r\) units away from the origin. Then, rotate around the origin by the angle \(\theta\). Any point can be represented in this manner, showcasing just how powerful and flexible this system can be.

Equivalent Equations

Equivalent equations in the context of polar and rectangular coordinates refer to different mathematical expressions that represent the same graph. For example, the rectangular equation \( x^2 + y^2 = 5y \) has an equivalent polar form of \( r = 5\sin\theta \). Despite these equations looking different, they describe exactly the same set of points or the graph.
When transforming an equation, check for equivalency by ensuring that both forms map out identical points. In our example, whether you plot the\( (x, y) \) points from the rectangular equation or the \( (r, \theta) \) points from the polar equation, the shape you get will remain the same.
Understanding how to determine and verify equivalency between different forms allows for greater flexibility in problem-solving and aids in choosing the most advantageous form for a particular problem. This skill is important for mathematical modeling, where different systems may highlight various aspects of the same underlying problem.