Question

Give two pairs of parametric equations that generate a circle centered at the origin with radius 6.

Short answer

Question: Find two pairs of parametric equations that generate a circle centered at the origin with a radius of 6. Answer: - First pair: x(t) = 6 * cos(t) and y(t) = 6 * sin(t) - Second pair: x(u) = 6 * cos(pi/2 - u) and y(u) = 6 * sin(pi/2 - u)

Step-by-step solution

First parametric equations

x(t) = 6 * cos(t) and y(t) = 6 * sin(t) Now, we need to find the second pair of parametric equations. One way to achieve this is by applying a trigonometric identity: cos(t) = cos(pi/2 - t) and sin(t) = sin(pi/2 - t)

Applying the trigonometric identity

Let u = pi/2 - t, then t = pi/2 - u. Substitute this into the first pair of parametric equations: x(u) = 6 * cos(pi/2 - u) y(u) = 6 * sin(pi/2 - u) Here is the second pair of parametric equations:

Second parametric equations

x(u) = 6 * cos(pi/2 - u) and y(u) = 6 * sin(pi/2 - u) So, the two pairs of parametric equations that generate a circle centered at the origin with radius 6 are: - First pair: x(t) = 6 * cos(t) and y(t) = 6 * sin(t) - Second pair: x(u) = 6 * cos(pi/2 - u) and y(u) = 6 * sin(pi/2 - u).

Key concepts

Circle Equations

A circle is a set of all points in a plane that are at a constant distance, called the radius, from a fixed point, called the center. The standard equation of a circle centered at the origin \((0, 0)\) with radius \(r\) is given by:\[x^2 + y^2 = r^2\]This equation encapsulates all points \((x, y)\) that lie on the circle. In this form, it's easy to see how the radius impacts the size of the circle. The larger the radius, the larger the circle, and vice versa.
The equation also highlights the symmetry of the circle—the perfect balance around its center point, which is crucial for various geometric applications.

Trigonometric Identities

Trigonometric identities are mathematical properties that apply to trigonometric functions like sine and cosine. These identities often express relationships between angles and can simplify solving problems.
One useful trigonometric identity is the complementary angle identity:- \(\cos(t) = \sin(\pi/2 - t)\)- \(\sin(t) = \cos(\pi/2 - t)\)These identities allow us to transform one trigonometric function into another based on different angles. They are particularly useful when defining parametric equations for geometric shapes. They help incorporate different paths or perspectives of the same shape without changing its properties.
These transformations can help find equivalent parametric representations, such as generating different circles from a basic equation.

Parametric Representation

Parametric representation is a way of describing geometric shapes using parameters rather than traditional coordinate equations. In this context, a parameter like \(t\) is used to define the coordinates \((x, y)\) of points on a circle.
For a circle centered at the origin, the parametric form can be:

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

These equations give the coordinates \((x, y)\) based on the angle \(t\), effectively tracing the circle's path as \(t\) varies from 0 to \(2\pi\).
Parametric equations provide flexibility and are often used in computer graphics and physics simulations to model various natural and man-made phenomena. By adjusting the parameter \(t\), you can dynamically explore all points along the circumference of the circle, which is a powerful way to visualize and manipulate shapes.

Radius of Circle

The radius of a circle is the distance from its center to any point on its boundary. This measurement plays a crucial role in defining a circle's size and its equation.
In the parametric equations \(x(t) = r \cdot \cos(t)\) and \(y(t) = r \cdot \sin(t)\), the factor \(r\) represents the circle's radius. In our specific problem, the radius is 6, as seen in the equations \(x(t) = 6 \cdot \cos(t)\) and \(y(t) = 6 \cdot \sin(t)\).
The radius can affect both the geometric and spatial properties of the circle. A larger radius results in a wider circle, while a smaller radius produces a more compact circle. By manipulating this simple measure, we can change how a circle appears, while preserving its essential circular properties. Understanding the role of the radius helps when designing or analyzing mechanical systems, where precision and consistency are vital.