Question

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

Short answer

Question: Provide two pairs of parametric equations that describe a circle centered at the origin with a radius of 6. Answer: 1. x(t) = 6 * cos(t), y(t) = 6 * sin(t) 2. x(u) = 6 / cos(u), y(u) = 6 / sin(u)

Step-by-step solution

Write the standard form of circle equation

The equation of a circle with center at the origin (0,0) and radius r can be written as: x^2 + y^2 = r^2 Since the given radius is 6, the equation of the circle would be: x^2 + y^2 = 36

Convert the equation to first parametric equation

We're going to use the trigonometric functions sine and cosine to create a parametric equation for the circle. Let t be an arbitrary parameter, and set x and y as follows: x = 6 * cos(t) y = 6 * sin(t) So, the first pair of parametric equations is: x(t) = 6 * cos(t) y(t) = 6 * sin(t)

Convert the equation to second parametric equation

For the second pair of parametric equations, we will use the trigonometric functions secant and cosecant. Let u be another arbitrary parameter, and set x and y as follows: x = 6 * sec(u) y = 6 * csc(u) Since sec(u) = 1/cos(u) and csc(u) = 1/sin(u), we can write x and y as: x = 6 / cos(u) y = 6 / sin(u) So, the second pair of parametric equations is: x(u) = 6 / cos(u) y(u) = 6 / sin(u) That's it! Here are the two pairs of parametric equations: 1. x(t) = 6 * cos(t), y(t) = 6 * sin(t) 2. x(u) = 6 / cos(u), y(u) = 6 / sin(u)

Key concepts

Circle Standard Form Equation

The standard form equation for a circle is a foundational concept in geometry, representing all points that are a fixed distance (the radius) away from a central point (the center). The equation for a circle with center at the origin (0,0) and radius r is expressed as
\( x^2 + y^2 = r^2 \).
This tidy formula emerges from the Pythagorean theorem and describes a perfect circle in Cartesian coordinates. In the instance where we have a circle with radius 6 centered at the origin, the equation becomes
\( x^2 + y^2 = 36 \),
encapsulating the set of all points (x, y) whose distance from the origin is 6. Understanding this formula is key to unlocking the mysteries of circular motion and patterns in algebra and calculus.

Trigonometric Functions

Trigonometric functions are the bridge between angles and side lengths in right-angled triangles, but they transcend their geometric origins to become indispensable tools in various mathematical fields.

Understanding Sine and Cosine

The functions sine (sin) and cosine (cos) relate an angle with the ratios of side lengths in a right triangle. Specifically, for an angle t, the sine function gives the ratio of the opposite side to the hypotenuse, while the cosine provides the adjacent side to the hypotenuse ratio.

Exploring Secant and Cosecant

The secant (sec) and cosecant (csc) are less commonly mentioned but are simply the reciprocals of cosine and sine, respectively. Therefore, sec(t) is the reciprocal of cos(t), and csc(t) is the reciprocal of sin(t). These functions play significant roles in constructing alternative parametric equations for a circle.

Parametric Equations for Circle

Parametric equations offer a dynamic way to represent curves by expressing coordinates as functions of one or more independent variables, generally time. When it comes to circles, parametric expressions shine by describing circular motion.

The First Pair of Equations

For a circle with radius 6 centered at the origin, one can use parametric equations to map this circle using trigonometric functions. By setting
\(x(t) = 6 \cdot \cos(t)\)
and
\(y(t) = 6 \cdot \sin(t)\),
every point (x(t), y(t)) on the circle corresponds to a specific value of t, representing an angle in radians.

The Second Pair of Equations

An alternative but equivalent set of parametric equations uses the secant and cosecant functions:
\(x(u) = 6 / \cos(u)\)
and
\(y(u) = 6 / \sin(u)\).
These offer another perspective of the same circle, highlighting the utility and flexibility of parametric representations. By grasping these equations, the patterns of circular motion become far more intuitive.

Circle with Center at Origin

A circle centered at the origin stands as a central figure in Cartesian coordinate systems. Its simpleness marks it as an ideal subject for exploring equations of circles and their parametric counterparts.

The Geometric Significance

When the center of a circle is positioned at the origin, any point on the perimeter is equally distanced from the center, defined by the radius. This symmetry makes calculations and transformations—including scaling, rotating, and reflecting—more straightforward mathematically.

Parametric Advantages

In parametric terms, a circle at the origin allows clear visualizations of trigonometric functions as they sweep through angles, creating an interconnected dance of (x,y) pairs tracing out the circle's path. This central location simplifies parametrizations since standard trigonometric functions—sine and cosine—natrually align with the circle's dimensions, providing an excellent tool for visualizing and solving problems related to circular shapes and motion.