Problem 63 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(53-66\) with equivalent polar equations. $$x^{2}+(y-2)^{2}=4$$

Step-by-Step Solution

Answer

The equivalent polar equation is \(r = 4 \sin(\theta)\).

Step 1: Convert Equation to Polar Formulas

First, recall the relationships between Cartesian coordinates \(x, y\)\ and their polar counterparts \((r, \theta)\)\:- \(x = r \cos(\theta)\)- \(y = r \sin(\theta)\).Replace \(x\) and \(y\) in the Cartesian equation \(x^{2} + (y-2)^{2} = 4\) with their polar forms.

Step 2: Substitute Polar Expressions

Substitute \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) into the original, giving \(\left(r \cos(\theta)\right)^2 + (r \sin(\theta) - 2)^2 = 4\).

Step 3: Expand and Simplify

Expanding the equation, we have:\((r \cos(\theta))^2 + (r \sin(\theta) - 2)^2 = 4\)\(r^2 \cos^2(\theta) + (r^2 \sin^2(\theta) - 4r \sin(\theta) + 4) = 4\).Simplify using \(r^2 \cos^2(\theta) + r^2 \sin^2(\theta) = r^2\), so:\(r^2 - 4r \sin(\theta) + 4 = 4\).

Step 4: Solve for r

Discuss the equation:Cancel out the \(4\) from both sides to get:\(r^2 - 4r \sin(\theta) = 0\).Factor the equation:\(r(r - 4 \sin(\theta)) = 0\).This gives possible solutions:\(r = 0\) or \(r = 4 \sin(\theta)\).

Key Concepts

Cartesian coordinates

In mathematics, Cartesian coordinates are crucial. They form a system that enables us to pinpoint the position of any point in a plane using two numerical values, usually marked as \(x\) and \(y\). This method allows for mapping points in either two-dimensional or three-dimensional space, providing a clear and comprehensive way to visualize points and shapes.

The "Cartesian" name originates from the French mathematician René Descartes, who developed this system in the 17th century. In our everyday math class, Cartesian coordinates help clarify the understanding of more complex concepts by giving a visual grounding for transformations and equations.

• In a 2D plane, \(x\) and \(y\) values indicate how far a point is from two perpendicular axes: the horizontal \(x\)-axis and vertical \(y\)-axis.
• The main point where both axes meet is called the origin, labeled as \((0, 0)\).
• Through this format, you can graph functions, draw shapes, and solve geometric problems easily.

Understanding Cartesian coordinates is fundamental not only for basic geometry but also for connecting to other mathematical systems, like polar coordinates. These relationships make it easier to convert solutions between different formats.

polar equations

Polar equations represent relations involving angles and distances from a central point, vastly differing from their Cartesian counterparts. In the polar system, each point on a plane is described by a pair, specifically: the radial distance \(r\) and the angular coordinate \(\theta\).

This alternate perspective can sometimes simplify equations or situate them in a more natural form for certain types of problems, such as those involving circles or periodic functions.

• \(r\) indicates the distance from the origin to the point, acting as a radius.
• \(\theta\) specifies the angle in radians measured from the positive \(x\)-axis to the radial line.
• These coordinates are particularly useful in fields of engineering and physics where systems rotate around a central point.

For instance, a circle traditionally expressed in Cartesian form may appear more straightforward in polar form, enhancing our perspective of its properties. Here, instead of troubling with the equation of a circle \(x^2 + (y - 2)^2 = 4\), transforming to polar gives us \(r = 4 \sin(\theta)\), demonstrating ease and clarity in its structure.

coordinate transformations

Coordinate transformations, like converting between Cartesian and polar systems, are pivotal in both mathematics and applied fields. These transformations are essentially decoding these systems into one another. It's like translating a language; each offers unique ways to express the same reality.

In transformations, you often leverage key relationships between Cartesian and polar coordinates:

• Use \(x = r \cos(\theta)\) to move from polar to Cartesian.
• Employ \(y = r \sin(\theta)\) for similar conversions.
• From Cartesian to polar, you can calculate \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(\frac{y}{x})\).

These steps guide the conversion process, granting the ability to translate complex systems into simpler forms or adapt calculations for specific applications. The Cartesian equation \(x^2 + (y-2)^2 = 4\) was succinctly converted into a polar form \(r = 4 \sin(\theta)\) by employing these transformation processes.

Mastering these transformations empowers individuals to tackle a variety of analytical problems, enriching both theoretical understanding and practical implementation.