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Magazine Chapter 6: Problem 9
Graph the plane curve for each pair of parametric equations by plotting points, and indicate the orientation on your graph using arrows. $$ x=\sin t, y=\cos t $$
Short Answer
Expert verified
The curve is a unit circle with a counter-clockwise orientation, centered at the origin.
Step by step solution
01
Understand the Parametric Equations
The given parametric equations are \( x = \sin t \) and \( y = \cos t \). These equations describe a plane curve where \( t \) is the parameter, often representing time or angle. We aim to plot the curve by evaluating these expressions for different values of \( t \).
02
Identify Range for Parameter t
Since both sine and cosine functions complete one full cycle over the interval \( [0, 2\pi] \), choose \( t \) values from \( 0 \) to \( 2\pi \) to plot the complete cycle of the curve. This will help us trace the full path of the curve on the plane.
03
Compute Points for Various t Values
Calculate \( x \) and \( y \) for a selection of \( t \) values within \( [0, 2\pi] \): - \( t = 0 \): \( x = \sin 0 = 0 \), \( y = \cos 0 = 1 \)- \( t = \frac{\pi}{2} \): \( x = \sin \frac{\pi}{2} = 1 \), \( y = \cos \frac{\pi}{2} = 0 \)- \( t = \pi \): \( x = \sin \pi = 0 \), \( y = \cos \pi = -1 \)- \( t = \frac{3\pi}{2} \): \( x = \sin \frac{3\pi}{2} = -1 \), \( y = \cos \frac{3\pi}{2} = 0 \)- \( t = 2\pi \): \( x = \sin 2\pi = 0 \), \( y = \cos 2\pi = 1 \)
04
Plot Computed Points
Plot the calculated points: - (0, 1) for \( t = 0 \)- (1, 0) for \( t = \frac{\pi}{2} \)- (0, -1) for \( t = \pi \)- (-1, 0) for \( t = \frac{3\pi}{2} \)- (0, 1) for \( t = 2\pi \).
05
Sketch the Curve Through the Points
Connect these points with a smooth curve. Notice that these points form a circle, known as the unit circle, centered at the origin with a radius of 1.
06
Indicate Orientation on the Graph
The direction indicated by increasing \( t \) (i.e., as \( t \) progresses from \( 0 \) to \( 2\pi \)) corresponds to moving counter-clockwise around the circle. Draw arrows on the curve pointing in the counter-clockwise direction to depict this orientation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Trigonometric Functions in Parametric Equations
Trigonometric functions, namely sine and cosine, play a crucial role in plotting parametric equations. In the given exercise, the parametric equations are defined as \( x = \sin t \) and \( y = \cos t \). These functions are periodic and oscillate between -1 and 1, repeating every full cycle from \( 0 \) to \( 2\pi \). This means:
- Sine function (\( \sin t \)) starts from 0, rises to 1 at \( \frac{\pi}{2} \), falls back to 0 at \( \pi \), descends to -1 at \( \frac{3\pi}{2} \), and comes back to 0 at \( 2\pi \).
- Cosine function (\( \cos t \)) begins at 1, decreases to 0 at \( \frac{\pi}{2} \), drops to -1 at \( \pi \), rises back to 0 at \( \frac{3\pi}{2} \), and ends at 1 at \( 2\pi \).
Exploring the Unit Circle
The unit circle is a foundational concept in trigonometry, providing a graphical representation of trigonometric functions. It is defined as a circle with a radius of 1 centered at the origin (0,0). When plotting \( x = \sin t \) and \( y = \cos t \), these coordinates trace the path of the unit circle.
- Each angle \( t \) on the unit circle corresponds to a point \((x, y)\) where \( x = \sin t \) and \( y = \cos t \).
- The cycle from \( 0 \) to \( 2\pi \) covers a full revolution, essentially tracing the whole circle.
Plotting Points and Understanding Orientation
Plotting points from parametric equations requires substituting various values of \( t \) to get the corresponding \( x \) and \( y \) coordinates. For the curve described by \( x = \sin t \) and \( y = \cos t \):
- For \( t = 0 \), point is (0, 1).
- For \( t = \frac{\pi}{2} \), point is (1, 0).
- For \( t = \pi \), point is (0, -1).
- For \( t = \frac{3\pi}{2} \), point is (-1, 0).
- For \( t = 2\pi \), point returns to (0, 1).
