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Chapter 6: Problem 1

Graph the plane curve for each pair of parametric equations by plotting points, and indicate the orientation on your graph using arrows. $$ x=3 \cos t, y=3 \sin t $$

Short Answer

Expert verified
The curve is a circle centered at the origin with radius 3, oriented counter-clockwise.

Step by step solution

01

Understand the Parametric Equations

The given parametric equations are \( x = 3 \cos t \) and \( y = 3 \sin t \). These equations describe a plane curve, where \( t \) is a parameter, typically representing an angle in radians.
02

Identify the Type of Curve

Notice that \( x = 3 \cos t \) and \( y = 3 \sin t \) resemble the parametric form of a circle. By recalling the identity \( \cos^2 t + \sin^2 t = 1 \), we can see this is a circle centered at the origin with radius 3.
03

Determine Key Points on the Curve

To graph the curve, choose values of \( t \), compute corresponding \( x \) and \( y \) values, and plot these points. For example: - \( t = 0 \), \( x = 3 \), \( y = 0 \)- \( t = \frac{\pi}{2} \), \( x = 0 \), \( y = 3 \)- \( t = \pi \), \( x = -3 \), \( y = 0 \)- \( t = \frac{3\pi}{2} \), \( x = 0 \), \( y = -3 \)- \( t = 2\pi \), \( x = 3 \), \( y = 0 \)
04

Plot the Points and Draw the Curve

Plot the points you calculated from Step 3 on a coordinate plane. Connect these points with a smooth, continuous curve to form a circle.
05

Indicate the Orientation

Since \( t \) increases, the orientation (or direction) of the curve is counter-clockwise. Indicate this by drawing arrows along the circle in a counter-clockwise direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circle Graphing
When working with parametric equations like \( x = 3 \cos t \) and \( y = 3 \sin t \), an understanding of circle graphing is essential. Circle graphing involves plotting a series of points derived from equations that represent the circumference of the circle.
These parametric equations describe a circle centered at the origin with a radius of 3. Why? Because they are directly linked to the trigonometric identity \( \cos^2 t + \sin^2 t = 1 \).
Here's how it works:
  • The radius, 3, is derived from the coefficients of \( \cos t \) and \( \sin t \), both of which are multiplied by 3.
  • The circle is centered at the origin \((0, 0)\) since there are no additional terms added to \( x \) or \( y \).
  • This configuration represents a standard way to circle graph, making it highly predictable and systematic.
Understanding circle graphing simplifies identifying and plotting points as it allows you to predict the circle's shape and orientation.
Trigonometric Functions
The use of trigonometric functions \( \cos t \) and \( \sin t \) in parametric equations is fundamental to describe the geometric path of a circle.
Trigonometric functions help express curves and cycles smoothly, thanks to their periodic nature. Here鈥檚 how they contribute:
  • \( \cos t \) represents the horizontal displacement. It oscillates between -1 and 1 as \( t \) increases, thus scaling horizontally by 3 in this instance, accounting for the circle's width.
  • \( \sin t \) represents the vertical displacement. Similarly, it also oscillates between -1 and 1, scaled by the radius here too, allowing for movement vertically.
  • Both functions complete their cycles every \( 2\pi \) radians, making them perfect for modelling circular paths since one full cycle traces a complete circle.
These trigonometric components ensure the path of the circle is not only complete but also smooth and continuously traced when used in parametric equations.
Plotting Points
Plotting points when dealing with parametric equations is key to constructing curves like a circle accurately. This process involves a few straightforward steps:
You begin by selecting key values of \( t \), which acts as your parameter:
  • Start with \( t = 0 \). Calculate \( x = 3 \cos(0) = 3 \) and \( y = 3 \sin(0) = 0 \). This gives the point (3, 0).
  • For \( t = \frac{\pi}{2} \), calculate \( x = 3 \cos(\frac{\pi}{2}) = 0 \) and \( y = 3 \sin(\frac{\pi}{2}) = 3 \). This yields the point (0, 3).
  • Continue this method for \( t = \pi \), \( t = \frac{3\pi}{2} \), and \( t = 2\pi \) to complete the circle with points (-3, 0), (0, -3), and (3, 0), respectively.
Plotting these calculated points and connecting them forms a continuous, smooth circular path.
To ensure that the circle is correctly traced, label and follow the orientation using arrows, typically in a counter-clockwise direction as you plot.

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Most popular questions from this chapter

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