Problem 56 - Algebra and Trigonometry

Question

Convert each rectangular equation to a polar equation that expresses r in terms of \(\theta\). $$ x^{2}+(y+3)^{2}=9 $$

Step-by-Step Solution

Answer

The polar equation obtained by conversion is \(r^{2}+6r \sin \theta = 0\).

Step 1: Substitute x and y

In the given equation \(x^{2}+(y+3)^{2}=9\), replace x with \(r \cos \theta\) and y with \(r \sin \theta\). This will yield \((r \cos \theta)^{2}+((r \sin \theta)+3)^{2}=9\).

Step 2: Simplify

Expand the squared brackets and group the terms. The resulting equation is \(r^{2} \cos^{2} \theta + r^{2}\sin^{2}\theta + 6r \sin \theta +9 = 9\).

Step 3: Further Simplification

Notice that \(r^{2} \cos^{2} \theta + r^{2}\sin^{2}\theta\) can be simplified to \(r^2\) because \(\cos^{2} \theta +\sin^{2}\theta = 1\), using the Pythagorean trigonometric identity. The resulting equation should then be \(r^{2}+6r \sin \theta +9 = 9\).

Step 4: Final Answer

The final equation can be obtained by subtracting 9 from both sides to achieve \(r^{2}+6r \sin \theta = 0\). This is the desired polar equation.

Key Concepts

Rectangular Equation Conversion

Converting rectangular equations to polar equations is a technique that allows us to represent equations in a coordinate system that may be more convenient for certain types of problems. The process involves utilizing the relationships between the rectangular coordinates

• x and y - the Cartesian coordinates representing horizontal and vertical positions.
• r and \(\theta\) - the polar coordinates where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis.

To convert from rectangular to polar, replace \(x\) with \(r \cos \theta\) and \(y\) with \(r \sin \theta\). This converts equations based on x and y to those based on r and \(\theta\). In the given exercise, we substitute these identities into the equation \(x^2 + (y+3)^2 = 9\) to get \[(r \cos \theta)^2 + ((r \sin \theta) + 3)^2 = 9\]This sets us on the path to rewrite the equation in terms of polar coordinates exclusively.

Trigonometric Identities

Trigonometric identities are fundamental tools that simplify the manipulation and transformation of equations involving trigonometric functions. In the conversion process to polar coordinates, one particular identity becomes extremely useful: the Pythagorean identity. This identity is expressed as \[\cos^{2} \theta + \sin^{2} \theta = 1\]This identity allows us to simplify expressions like \(r^{2} \cos^{2} \theta + r^{2} \sin^{2} \theta\) to just \(r^2\). Applying this identity during the simplification step of our conversion process, we obtain \[r^{2} \cos^{2} \theta + r^{2} \sin^{2} \theta + 6r \sin \theta + 9 = 9\] which becomes \[r^{2} + 6r \sin \theta + 9 = 9\]By recognizing and applying these identities, we streamline complex expressions into simpler forms that are easier to interpret and analyze.

Coordinate Systems

Understanding different types of coordinate systems is essential in mathematics for solving diverse problems effectively. Here, the focus is on two specific types:

• Rectangular coordinates (Cartesian): These represent points through x and y values within a two-dimensional plane. This system delineates positions with straight horizontal and vertical lines.
• Polar coordinates: In polar coordinates, each point is defined by a distance from a reference point (origin) and an angle from a reference direction (positive x-axis). It’s particularly useful for circular and angular problems.

Switching between these systems can clarify certain properties of equations. For example, circular behavior is often described more naturally in polar form, which deals explicitly with radii and angles. The procedural conversion from rectangular to polar coordinates, as showcased in this exercise, provides a methodical approach to translate an equation into a potentially more insightful format. This conversion allows us to explore geometric and rotational symmetries that might not be evident in the Cartesian form.