Problem 59 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(49-62\) by equivalent polar equations. $$ x^{2}+(y-2)^{2}=4 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

The given equation is a circle in Cartesian coordinates: \(x^2 + (y-2)^2 = 4\). The circle is centered at (0, 2) and has a radius of 2.

Step 2: Convert Cartesian to Polar Coordinates

We use the relationship between Cartesian and polar coordinates, where \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Substitute these into the Cartesian equation.

Step 3: Substitute Polar Expressions

Replace \(x\) and \(y\) in the equation: \(x^2 + (y-2)^2 = 4\) becomes \( (r\cos\theta)^2 + (r\sin\theta - 2)^2 = 4\).

Step 4: Simplify the Expression

Expand and simplify the equation: \(r^2\cos^2\theta + (r\sin\theta - 2)^2 = 4\) becomes \(r^2\cos^2\theta + r^2\sin^2\theta - 4r\sin\theta + 4 = 4\).

Step 5: Combine Like Terms

Notice that \(r^2\cos^2\theta + r^2\sin^2\theta = r^2\) due to the Pythagorean identity. Replace and further simplify: \(r^2 - 4r\sin\theta + 4 = 4\).

Step 6: Isolate \(r^2\)

Subtract 4 from both sides of the equation: \(r^2 - 4r\sin\theta = 0\).

Step 7: Factor and Solve for \(r\)

Factoring gives: \(r(r - 4\sin\theta) = 0\). This implies \(r = 0\) or \(r = 4\sin\theta\). Since \(r = 0\) corresponds to the origin, the polar equation is \(r = 4\sin\theta\).

Key Concepts

Cartesian Coordinates

You'll often see problems in math that start with Cartesian coordinates. This system uses a grid-based approach, with two axes: the x-axis (horizontal) and the y-axis (vertical). Every point on this plane is defined by both an x-value and a y-value, written as (x, y). This allows us to pinpoint where each point is located pretty easily.
For example, the Cartesian equation given to us is \(x^2 + (y-2)^2 = 4\), which describes a circle. In this equation:

• The circle is centered at the point (0, 2).
• The radius of the circle is 2.

Being able to understand Cartesian coordinates and the shapes they represent, like circles, is very important. This helps us make complex transformations into other coordinate systems easier to grasp.

Coordinate Conversion

One concept that often pops up in math is how to switch from one type of coordinates to another—known as coordinate conversion. In this exercise, we move from Cartesian to polar coordinates. Instead of x-y pairs, polar coordinates use a distance and an angle.
The key formulas to remember are:

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)

Using these formulas, you can translate any point from Cartesian coordinates to polar coordinates. In our equation, we replace the x and y terms with expressions involving r (the radius in polar coordinates) and \(\theta\) (the angle). This gives us a new equation more suitable for understanding in polar terms.

Circle Equations

Circle equations are easier to understand once you get the hang of them. In Cartesian coordinates, a circle's equation can be derived using the Pythagorean theorem. The standard form is \((x - h)^2 + (y - k)^2 = r^2\), where (h, k) is the center and r is the radius.
But things are different when you convert this into a polar equation. In our solution, we got to the step \(r = 4\sin(\theta)\). This polar equation tells us the same information as the Cartesian one, just from a different perspective:

• The circle is centered around a line, specifically when \(r\) changes depending on \(\theta\).
• The values of \(r = 4\sin(\theta)\) create the circle we described initially.

Both coordinate systems help us visualize the same circle, even if they use different methods. Understanding both opens up more tools for solving complex mathematical problems.