Question

Find parametric equations (not unique) for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of \(x\) and \(y\) A circle centered at the origin with radius \(4,\) generated counterclockwise

Short answer

Question: Provide the parametric equations for a circle centered at the origin with radius 4 generated counterclockwise, the interval of parameter values, and the description of the circle in terms of x and y. Answer: The parametric equations for the circle are \(x(t) = 4 \cos{t}\) and \(y(t) = 4 \sin{t}\). The parameter values interval is \(0 \leq t \leq 2\pi\) (or \(0^\circ \leq t \leq 360^\circ\) in degrees). The description of the circle in terms of x and y is \(x^2 + y^2 = 16\).

Step-by-step solution

Find Parametric Equations

For a circle with radius \(r\) and generated counterclockwise, the parametric equations are given by: \(x = r \cos{t}\) \(y = r \sin{t}\) As the circle has a radius of \(4\) and is centered at the origin, the parametric equations are: \(x = 4 \cos{t}\) \(y = 4 \sin{t}\) So, the parametric equations for this circle are: \(x(t) = 4 \cos{t}\) \(y(t) = 4 \sin{t}\)

Determine the Interval for the Parameter Values

Since we want the circle generated counterclockwise, we need to look at the interval of the parameter \(t\) in which the trigonometric functions generate one complete revolution. For the sine and cosine functions, this interval in radians is \(0 \leq t \leq 2\pi\). If we prefer degrees, the interval is \(0^\circ \leq t \leq 360^\circ\).

Find a Description in Terms of x and y

To find the description of the circle in terms of \(x\) and \(y\), we can simply eliminate the parameter \(t\) from the parametric equations and express the relation between \(x\) and \(y\) only. We can do this by squaring both parametric equations and adding them: \((x(t))^2 = (4 \cos{t})^2\) \((y(t))^2 = (4 \sin{t})^2\) Now, add the left sides together and the right sides together: \((x(t))^2 + (y(t))^2 = (4 \cos{t})^2 + (4 \sin{t})^2\) Apply the Pythagorean identity (\(\cos^2{t} + \sin^2{t} = 1\)): \((x(t))^2 + (y(t))^2 = (4^2)(\cos^2{t} + \sin^2{t}) = 16\) So the description of the circle in terms of \(x\) and \(y\) is: \(x^2 + y^2 = 16\) In conclusion, the parametric equations for the circle are \(x(t) = 4 \cos{t}\) and \(y(t) = 4 \sin{t}\) with the parameter values interval being \(0 \leq t \leq 2\pi\) (or \(0^\circ \leq t \leq 360^\circ\) in degrees), and the description in terms of \(x\) and \(y\) is \(x^2 + y^2 = 16\).

Key concepts

Circle Equation

The equation of a circle is an elegant yet fundamental concept in mathematics.Understanding the equation helps us represent the circle's properties mathematically.The general form of a circle's equation in a two-dimensional plane is \((x - h)^2 + (y - k)^2 = r^2\). Here:

• \((h, k)\) are the coordinates of the circle's center.
• \(r\) represents the radius of the circle.

In our specific problem, the circle is centered at the origin \((0,0)\) and has a radius of 4, simplifying the circle equation to \(x^2 + y^2 = 16\).This form is derived from substituting \(h = 0\), \(k = 0\), and \(r = 4\).Thus, the circle's equation succinctly captures its position and size.

Trigonometric Functions

Trigonometric functions are crucial when dealing with circular motion, as they provide a way to relate angles to coordinates on a unit circle.In the context of this circle, the parametrization using these functions is centered on understanding how cosine and sine define the circle's shape.

• \(x(t) = 4 \cos{t}\)
• \(y(t) = 4 \sin{t}\)

These parametric equations map the angle \(t\) to a point on the circle.Cosine and sine repeat every \(2\pi\) radians (or 360 degrees), which corresponds to completing one full loop around the circle.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows mathematicians to represent geometric shapes numerically and analyze their properties through algebraic equations.In our problem, coordinate geometry is used to build parametric equations that describe a circle centered at the origin with a radius of 4.These equations aren’t just abstract formulations; they connect the geometry of circles with algebra through parameters, aiding both visualization and computation. By setting specific values for \(t\), one can compute exact points on the circle, seamlessly linking geometric intuition with algebraic manipulation.

Radius and Center of Circle

The radius and center are two fundamental components that define a circle's geometry.

• The **radius** is the distance from the center of the circle to any point on its circumference.
• The **center** is a fixed internal point equidistant from all points on the circle's outline.

In our scenario, the circle is centered at the origin (\(0,0\)), a convenient choice simplifying calculations.The stated radius is 4, determining the circle's size and stretching it equidistantly in every direction from the center.This uniform distance implies that despite the circle's rotation or movement, every point remains within this specific reach, maintaining perfect symmetry and balance.