Problem 2 Graph the plane curve for each p... [FREE SOLUTION]

Chapter 6: Problem 2

Graph the plane curve for each pair of parametric equations by plotting points, and indicate the orientation on your graph using arrows. $$ x=2 \sin t, y=2 \cos t $$

Short Answer

Expert verified

The curve is a circle centered at the origin with radius 2, oriented counterclockwise.

Step by step solution

Identify the Parametric Equations

The given parametric equations are $ x = 2 \sin t $ and $ y = 2 \cos t $. These equations define the coordinates $ (x, y) $ in terms of the parameter $ t $.

Determine the Shape of the Curve

Recognize that if $ x = 2 \sin t $ and $ y = 2 \cos t $, then this can model a circle. This is a classic pattern, and these are trigonometric functions, hinting it could be a circular path.

Eliminate the Parameter

To verify the shape, eliminate the parameter $ t $. Use the identities $ \sin^2 t + \cos^2 t = 1 $ and substitute: $ \left( \frac{x}{2} \right)^2 + \left( \frac{y}{2} \right)^2 = \sin^2 t + \cos^2 t = 1 $. This simplifies to $ \frac{x^2}{4} + \frac{y^2}{4} = 1 $, which is the equation of a circle with radius 2.

Plot Points

Pick values for $ t $, calculate $ x $ and $ y $, and plot these points. For example: when $ t = 0 $, $ x = 0, y = 2 $; when $ t = \frac{\pi}{2} $, $ x = 2, y = 0 $; when $ t = \pi $, $ x = 0, y = -2 $; and when $ t = \frac{3\pi}{2} $, $ x = -2, y = 0 $.

Indicate Orientation

The parameter $ t $ starts at 0 and increases. Track the order you calculated points: $(0, 2)$, $(2, 0)$, $(0, -2)$, $(-2, 0)$. Draw arrows on the circle through these points to indicate a counterclockwise direction, confirming the orientation of the path.

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Key Concepts

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

Trigonometric Functions

Trigonometric functions, sine and cosine, play a critical role in understanding parametric equations. These functions relate to angle measures in a right triangle, with sine representing the ratio of the length of the side opposite the angle to the hypotenuse, and cosine representing the adjacent side's ratio to the hypotenuse.

The sine function varies between -1 and 1 as the angle changes from 0 to 360 degrees (or 0 to 2蟺 radians).
Similarly, the cosine function also varies between -1 and 1, but it starts at 1 when the angle is 0, creating a similar periodic wave.

These wave-like patterns make trigonometric functions ideal for plotting circular paths, as demonstrated in the parametric equations $x = 2 \sin t$ and $y = 2 \cos t$. Knowing these properties allows us to predict and verify the shape of the graph more easily.

Unit Circle

The unit circle is a fundamental concept when dealing with trigonometric functions. It is a circle with a radius of one, centered at the origin of a coordinate plane. By using the unit circle, we can extend the basic definitions of sine and cosine beyond right-angled triangles.

Each angle $ t $ on the unit circle corresponds to a certain point $(\cos t, \sin t)$.
The parametric equations in the exercise follow a similar structure to this, but are scaled by a factor of 2.
Given the equations $x = 2 \sin t$ and $y = 2 \cos t$, each of these values is doubled from their position on the unit circle. Therefore, the resulting circle has a radius of 2 instead of 1.

Understanding this scaling is key in moving from the theoretical aspect of the unit circle to practical applications in graphing parametric curves.

Graphing Parametric Curves

Graphing parametric curves involves tracking the movement of a point across a plane, guided by parametric equations. In this exercise, we derived a circle from the equations $x = 2 \sin t$ and $y = 2 \cos t$.

Firstly, we eliminated the parameter $t$ using identities like $\sin^2 t + \cos^2 t = 1$, which helped us confirm the shape as a circle.
By plotting points for various values of $t$, such as $t = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$, we observe a circular path.
Marking these points and drawing lines among them allows us to visualize the curve.
If we observe the order of these points, we can determine the direction of motion, which helps to indicate orientation with arrows.

This graphical representation is an effective method to visualize and understand the path defined by parametric equations.

91影视

Graph the plane curve for each pair of parametric equations by plotting points, and indicate the orientation on your graph using arrows. $$ x=2 \sin t, y=2 \cos t $$

Short Answer

Step by step solution

Identify the Parametric Equations

Determine the Shape of the Curve

Eliminate the Parameter

Plot Points

Indicate Orientation

Key Concepts

Trigonometric Functions

Unit Circle

Graphing Parametric Curves

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