Question

Describe how their graphs are similar and different. Be sure to discuss orientation and ranges. \(\quad\) (a) \(x=\cos t \quad y=\sin t, \quad 0 \leq t \leq 2 \pi\) (b) \(x=\cos \left(t^{2}\right) \quad y=\sin \left(t^{2}\right), \quad 0 \leq t \leq 2 \pi\) (c) \(x=\cos (1 / t) \quad y=\sin (1 / t), \quad 0

Short answer

Each graph is based on the unit circle but covers it differently due to transformations in their parameters.

Step-by-step solution

Analyze Graph (a)

The parametric equations \( x = \cos t \) and \( y = \sin t \) describe a circle with radius 1 centered at the origin. As \( t \) varies from 0 to \( 2\pi \), the graph traces the unit circle counterclockwise once. The range of \( x \) and \( y \) is \([-1, 1]\).

Analyze Graph (b)

For \( x = \cos(t^2) \) and \( y = \sin(t^2) \), the parametric graph has \( t^2 \) as the argument, so its range of angles will be denser closer to 0 and sparse near \( 2\pi \). This does not cover the entire unit circle evenly due to the squared term. The orientation remains counterclockwise, and the range is still \([-1, 1]\).

Analyze Graph (c)

The equations \( x = \cos(1/t) \) and \( y = \sin(1/t) \) cause the t values approaching 0 from the right (\(0 < t < 1\)) to result in increasing frequency oscillations. This forms a dense set of points near the origin with a complex spiral pattern. Both \( x \) and \( y \) still have a range of \([-1, 1]\) due to the nature of sine and cosine.

Analyze Graph (d)

In \( x = \cos(\cos t) \) and \( y = \sin(\cos t) \), the arguments for sine and cosine range between -1 and 1, rather than \( t \) covering \( 0 \) to \( 2\pi \). This results in the graph tracing arcs within the circle's first quadrant. The orientation remains counterclockwise, but it's incomplete and limited within the central location of the circle.

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and are widely used in the description of periodic behaviors. In the context of these parametric equations, cosine and sine play a key role. Each function, \( \cos(t)\) and \( \sin(t)\), oscillates between -1 and 1. This oscillating behavior is fundamental in cases like (a) where \( x = \cos(t)\) and \( y = \sin(t)\) describe a perfect circle. - \( \cos(t^2)\) and \( \sin(t^2)\) maintain this oscillation, but change in their density around the circle due to the squaring of \( t\).- \( \cos(1/t)\) and \( \sin(1/t)\) demonstrate a different application by introducing small values into the argument, resulting in a spiral effect.- Finally, \( \cos(\cos t)\) and \( \sin(\cos t)\) make use of a cosine function within its argument, concentrating values within the first quadrant. Understanding the behavior of \( \cos\) and \( \sin\) is crucial for predicting and plotting their trajectories in different scenarios.

Graph Orientation

Orientation in graphs built from parametric equations indicates the direction in which the curve is traced as the parameter varies. In the provided examples, orientation is described by the range and transformation of \( t\). - For graph (a), the parameter \( t\) increases from 0 to \( 2\pi\), causing the graph to trace in a smooth, counterclockwise direction on the unit circle, a classic scenario of trigonometric parametric plots.- In graph (b), where \( x = \cos(t^2)\) and \( y = \sin(t^2)\), the orientation remains generally counterclockwise, but the density shifts due to the \( t^2\) transformation.- Graph (c) exhibits unique behavior by \( x = \cos(1/t)\) and \( y = \sin(1/t)\), as small increments in \( t\) create high frequency oscillations forming a spiral close to the origin.- For graph (d), \( x = \cos(\cos t)\) and \( y = \sin(\cos t)\) trace arc patterns that maintain a counterclockwise orientation but are limited to the first quadrant, showcasing the transformation effects of trigonometric arguments. The orientation of each curve deeply affects its aesthetic and is a crucial part of analyzing parametric plots.

Range of Functions

The range of a function depicts the output values a function can take. In trigonometric parametric equations, determining the range involves understanding the properties of \( \cos\) and \( \sin\). - Graph (a), defined by \( x = \cos t\) and \( y = \sin t\), exhibits the range \([-1, 1]\) for both axes because the output of both functions naturally fluctuates between these limits.- Graph (b)'s \( x = \cos(t^2)\) and \( y = \sin(t^2)\) uphold the same range, \([-1, 1]\), since squaring \( t\) does not affect the eventual sine and cosine outcomes.- In case (c), \( x = \cos(1/t)\) and \( y = \sin(1/t)\) adhere to the range \([-1, 1]\) despite the oscillations and small \( t\) values because they still compute through sine and cosine.- For graph (d), the functions \( x = \cos(\cos t)\) and \( y = \sin(\cos t)\) also provide outputs in the range \([-1, 1]\), emphasizing how plugging trigonometric results back into trigonometric functions doesn't exceed this range. Evaluating the range is essential for verifying and understanding the behavior and extent of parametric equations in a graphical context.