Problem 15 - Essential Calculus

Question

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=3 \cos \theta, \quad y=3 \sin \theta $$

Step-by-Step Solution

Answer

The corresponding rectangular equation to the given parametric equations is \(x^2 + y^2 = 9\), which sketches a circle with its center at the origin, a radius of 3 units and is oriented counterclockwise starting at (3,0).

Step 1: Convert Parametric Equations to Rectangular Form

To transcribe the provided parametric equations to a rectangular form, the cosine and sine of \(\theta\) in them have to be squared and summed. This yields: \(x^2 + y^2 = (3\cos{\theta})^2 + (3\sin{\theta})^2 = 9\).

Step 2: Sketching the Curve

The rectangular equation \(x^2 + y^2 = 9\) is indicative of a circle with its center at origin (0,0) and a radius of 3 units. Draw a circle with this configuration. The orientation of the curve can be inferred from the parameter \(\theta\). As \(\theta\) increases from 0 to \(2\pi\), point (x, y) moves counterclockwise around the circle, starting and ending at (3, 0).

Step 3: Final Details

Affix the orientation of the curve to your sketch. Starting at the point (3, 0) and moving counterclockwise, overlay an arrow pointing towards the direction of travel. This highlights the path taken by (x, y) as \(\theta\) increases from 0 to \(2\pi\).

Key Concepts

Rectangular Equation

Understanding the bridge between parametric equations and their rectangular counterparts is essential in algebra and trigonometry. Parametric equations, such as x=3 \/cos \theta and y=3 \/sin \theta, provide a set of functions where each variable depends on an independent parameter, in this case, \( \theta \).

The task is to eliminate this parameter to find the 'rectangular equation'—a single equation relating x and y directly without a parameter. To achieve this, we leverage the Pythagorean identity \( \cos^2\theta + \sin^2\theta = 1 \). After squaring both parametric equations and adding them, we obtain \( x^2 + y^2 = 9 \), which is your standard circle equation x^2 + y^2 = r^2 with radius r being 3 units.

This process is immensely useful as it transforms the dynamic representation of movement (parametric form) into a stationary standard form (rectangular equation) that is often easier to visualize and work with algebraically.

Sketching Curves

When sketching curves, it can be helpful to understand the geometrical interpretation of equations. Rectangular equations like \( x^2 + y^2 = 9 \) describe a relationship between the x and y coordinates in a static, universally understood way.

To sketch the curve of the circle indicated by the equation \( x^2 + y^2 = 9 \), one can simply draw the circle with the center at the origin (0,0) and a radius of 3 units. Visualizing the result of the equation as a set of points on the Cartesian plane aids in comprehending the concept and helps reassure that the conversion from parametric to rectangular form was correct.

Drawing Step-by-Step

• Identify the type of shape: in this case, it's a circle.
• Determine the center: here, it's at the origin (0,0).
• Find the radius: from the equation, we know it's 3.
• Draw the circle using the radius, marking the center point.

Orientation of a Curve

The orientation of a curve is the direction which you trace the curve, and it lends an additional layer of information to the two-dimensional sketch. More than just showing a static image, this tells you which direction the object would travel along the path, were it dynamic.

In the given parametric equations, the orientation is derived from how the parameter \( \theta \) influences the coordinates (x,y). Since \( \theta \) increases counterclockwise from 0 to \( 2\pi \), it tells us that the curve (here, our circle) will also be traced in a counterclockwise fashion starting from the point (3, 0), which corresponds to \( \theta = 0 \). To depict this orientation on your sketch:

• Start at the point (3, 0) on the circle's circumference.
• Draw an arrow along the curve moving in the counterclockwise direction.
• This provides a visual cue of the direction in which the curve is traced as \( \theta \) increases.

Understanding orientation is particularly important in physics and engineering, where the direction of movement often has implications for the behavior of systems.