Problem 27 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The parametric equations of the circle are x = 4cosθ and y = 4sinθ, where the interval for parameter θ is 0 ≤ θ < 2π.

Step 1: Write the parametric equations for the circle

Use the parametric equations of a circle formula and substitute values \((h=0, k=0, r=4)\): $$ x = 4\cos{\theta}, $$ $$ y = 4\sin{\theta}. $$ These are the parametric equations for the given circle.

Step 2: Find the interval for the parameter values

Since \(\theta\) represents the angle in a counterclockwise motion around the circle, it ranges from \(0\) to \(2\pi\). The interval for the parameter values is \(0 \le \theta < 2\pi\).

Step 3: Graph the circle

To graph the circle, plot the points \((x, y)\) generated by the parametric equations as the parameter \(\theta\) varies from \(0\) to \(2\pi\). The graph will be a circle centered at the origin with a radius of \(4\).

Step 4: Find a description in terms of x and y

Eliminate the parameter \(\theta\) to find a description in terms of \(x\) and \(y\). We can use the following equations: $$ x = 4\cos{\theta}, $$ $$ y = 4\sin{\theta}. $$ Square both equations and add them: $$ x^2 = 16\cos^2{\theta}, $$ $$ y^2 = 16\sin^2{\theta}. $$ $$ x^2 + y^2 = 16\cos^2{\theta} + 16\sin^2{\theta}. $$ Factor out \(16\): $$ x^2 + y^2 = 16(\cos^2{\theta} + \sin^2{\theta}). $$ Since \(\cos^2{\theta} + \sin^2{\theta} = 1\), we have: $$ x^2 + y^2 = 16. $$ The description of the circle in terms of \(x\) and \(y\) is \(x^2 + y^2 = 16\).

Key Concepts

Parametric Equations

Parametric equations represent a set of related quantities as explicit functions of an independent variable, known as a parameter. In the context of the circle exercise, the parametric equations are a powerful way to describe the circle's position at any given time. By using the parameter \(\theta\), the angle in radians, we can express the coordinates of any point on the circle as \(x = r\cos{\theta}\) and \(y = r\sin{\theta}\) where \(r\) is the radius of the circle.

The beauty of parametric equations lies in their ability to capture motion and direction, making them especially useful for graphing shapes that rotate or move over time. In our example, the parameter \(\theta\) varies from \(0\) to \(2\pi\), enabling us to trace the entire circumference of the circle in a counterclockwise direction.

Graphing Circles

Graphing circles involves plotting the set of all points that are the same distance (the radius) from a fixed point (the center). The equation \(x^2 + y^2 = r^2\) represents a circle centered at the origin, where the radius is \(r\).

When using parametric equations to graph a circle, like in our exercise with a circle of radius \(4\), we generate points using the equations \(x = 4\cos{\theta}\) and \(y = 4\sin{\theta}\) and plot them as \(\theta\) varies. This process defines the shape incrementally and elegantly demonstrates the circular path on the coordinate plane. Notably, parametric equations also allow for the easy graphing of circles that are not centered at the origin by adjusting the constants in the equations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables they contain. One of the most fundamental identities you encounter when dealing with circles is the Pythagorean identity: \(\cos^2{\theta} + \sin^2{\theta} = 1\).

This identity is crucial when converting between the parametric equations and the standard form equation of a circle. In our exercise, we used the Pythagorean identity to eliminate the parameter \(\theta\) and show that the points \(x, y\) lying on the circle satisfy the equation \(x^2 + y^2 = 16\). Understanding these identities helps simplify many complex trigonometric expressions and is essential for solving geometry and calculus problems.

Polar Coordinates

Polar coordinates provide an alternative way of representing the location of a point in a plane using the distance from a reference point (the pole) and an angle from a reference direction. In the polar coordinate system, any point \(P\) can be represented as \( (r, \theta) \), where \(r\) is the radial distance from the pole and \(\theta\) is the polar angle.

Parametric equations for a circle can be seen as a special case of polar coordinates, with the center of the circle acting as the pole and the radius as the constant radial distance. By manipulating the parametric equations, as we did in the textbook exercise, we are indirectly working with polar concepts, which also offer a rich framework for studying curves and systems that exhibit radial symmetry.