Problem 35 - Precalculus with Limits

Question

In Exercises 33-40, use the results of Exercises 29-32 to find a set of parametric equations for the line or conic. Circle: center: \((3, 2);\) radius: \(4\)

Step-by-Step Solution

Answer

The set of parametric equations for the given circle is \(x = 3 + 4 \cos(\theta)\) and \(y = 2 + 4 \sin(\theta)\)

Step 1: Set up the equations

The standard form of the parametric equations for a circle centered at \((h, k)\) with radius \(r\) is given by: \(x = h + r \cos(\theta)\) and \(y = k+ r \sin(\theta)\). Here, \(\theta\) is a parameter. For the given circle, \(h = 3\), \(k = 2\), and \(r = 4\)

Step 2: Substitute values

Substitute \(h = 3\), \(k = 2\), and \(r = 4\) into the standard form equations. This gives us the parametric equations for the circle: \(x = 3 + 4 \cos(\theta)\) and \(y = 2 + 4 \sin(\theta)\)

Key Concepts

Circle Equations

Circle equations are fundamental in mathematics, especially when dealing with coordinate geometry. The standard equation for a circle, particularly in a Cartesian coordinate system, is typically written as \((x-h)^2 + (y-k)^2 = r^2\). This equation represents a circle with its center at the point \((h, k)\) and a radius of \(r\). When we translate this into parametric equations, we focus on expressing both \(x\) and \(y\) in terms of a single parameter, generally denoted as \(\theta\). This parameter represents the angle in the plane, measured from the positive x-axis.

• This approach allows us to describe any point on the circle using trigonometric functions.
• The parametric form is easy to manipulate and particularly useful for graphing and animations.

For the given scenario, substituting the center \((3, 2)\) and radius \(4\), we derived the parametric equations:

• \(x = 3 + 4 \cos(\theta)\)
• \(y = 2 + 4 \sin(\theta)\)

Trigonometric Functions

Trigonometric functions play a crucial role in forming parametric equations of circles. The most common trigonometric functions used in these contexts are sine and cosine. These functions help to express the coordinates in terms of a parameter that sweeps through angles. Here's why they are used:

• \(\cos(\theta)\) and \(\sin(\theta)\) are periodic functions. They allow us to describe continuous motion along the circular path.
• These functions have a range between -1 and 1, which perfectly fits the need to map a circle of unit radius and then scale to any size.

Using trigonometric functions in parametric equations not only simplifies our equations but also makes it easier to visualize and compute paths around a circle.
With a radius \(r\), any point \((x, y)\) on the circle can be described as:

• \(x = h + r \cos(\theta)\)
• \(y = k + r \sin(\theta)\)

These equations ensure that as \(\theta\) changes, the points smoothly trace the perimeter of the circle.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves the use of coordinates to investigate geometric shapes like circles. It connects algebra and geometry through the use of graphs and formulas. By using coordinate systems, we can analyze the properties of geometric figures in a straight-forward manner.
Here's how it applies to circles:

• We use coordinates \((x, y)\) to locate points on the plane.
• Equations are used to produce geometric shapes and understand their properties.

For circles, coordinate geometry helps in defining not just the shape, but also its location and size. The equations can further be altered for transformations such as translation and dilation without changing the fundamental properties of the circle.
In the context of parametric equations, we redefine the circle’s equation by incorporating trigonometric functions, using the circle's center and radius. This allows a seamless transition between geometric understanding and algebraic manipulation, which is extremely beneficial in various applications such as physics simulations and engineering designs.