Problem 31 - College Trigonometry

Question

In Exercises \(25-39\), find a parametric description for the given oriented curve. the circle \(x^{2}+y^{2}=25\), oriented counter-clockwise

Step-by-Step Solution

Answer

The parametric equations are \(x(t) = 5\cos(t)\), \(y(t) = 5\sin(t)\) for \(0 \leq t < 2\pi\).

Step 1: Identify the Key Aspects

First, recognize that the equation \(x^2 + y^2 = 25\) represents a circle centered at the origin with a radius of 5. We need a parametric form that describes this circle and adheres to the given orientation.

Step 2: Define Basic Parametric Equations for the Circle

Knowing that a standard parametric form for a circle centered at the origin is \(x = r\cos(t)\) and \(y = r\sin(t)\), where \(r\) is the radius, we can substitute \(r = 5\) for our specific circle.

Step 3: Include the Correct Orientation

The problem specifies a counter-clockwise orientation. The parametrization \(x = 5\cos(t)\) and \(y = 5\sin(t)\) with \(t\) ranging from 0 to \(2\pi\) ensures the counter-clockwise traverse around the circle.

Step 4: Combine into Complete Parametric Equations

The parametric description of the circle is: \(x(t) = 5\cos(t)\) and \(y(t) = 5\sin(t)\), where \(t\) varies from \(0\) to \(2\pi\).

Key Concepts

Circle Equations

Circles are one of the most fundamental geometric shapes encountered in mathematics. The standard form of a circle's equation is \(x^2 + y^2 = r^2\), where \(r\) represents the radius of the circle. For a circle centered at the origin \((0,0)\), the equation simplifies to this form where \(r\) is simply the distance from the center to any point on the circle.
In our exercise, the circle is defined by \(x^2 + y^2 = 25\). To determine the radius, we recognize that \(25\) is equal to \(r^2\). Therefore, taking the square root of both sides reveals that \(r\) is 5. This tells us that every point \((x, y)\) on the circle is exactly 5 units away from the origin.
Understanding these equations is crucial for defining paths and shapes in various fields: from navigation systems to graphical designs.

Trigonometric Parametrization

Parametric equations are an important concept when we want to describe a shape using parameters that may represent time, angle, or another variable. For a circle, trigonometric parametrization is often the most straightforward approach.
Consider the circle centered at the origin with a radius \(r\). The parametric equations for this circle are:

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

Here, \(t\) typically represents an angle in radians and is our parameter. As \(t\) varies from 0 to \(2\pi\), we trace out the circle counter-clockwise.
By substituting the radius \(r = 5\) from our specific problem, we obtain:

• \(x(t) = 5\cos(t)\)
• \(y(t) = 5\sin(t)\)

This gives us a complete parametric description ensuring we move counter-clockwise around the circle as \(t\) increases from 0 to \(2\pi\). The use of trigonometric functions, \(\cos\) and \(\sin\), captures the circular nature of the shape elegantly and efficiently.

Geometric Shapes and Orientations

In geometry, understanding both the shape and its orientation is key to fully describing an object. Orientation indicates how a curve or shape is being traversed, and it plays a crucial role when using parametric equations.
For the circle given by \(x^2 + y^2 = 25\), we need it to be described in a counter-clockwise manner. Trigonometric parametrization naturally facilitates this orientation. As \(t\) increases, the \(x\) and \(y\) coordinates evolve based on \(\cos(t)\) and \(\sin(t)\), ensuring a smooth counter-clockwise path.
Why does this matter? Orientation can affect calculations and applications such as animations or path-following algorithms. Knowing whether to follow a shape clockwise or counter-clockwise determines how the object or path is visually and practically interpreted.
Understanding these aspects ensures one correctly sets the stage for further mathematical exploration or practical implementations involving more complex shapes and patterns.