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Chapter 6: Problem 44

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=\sec \theta\\\ &y=\tan \theta \end{aligned}$$

Short Answer

Expert verified
Using a graphing utility, the graph will represent the right-hand side of a circle centered at the origin with a radius of 1 (not inclusive of zero).

Step by step solution

01

Understand the Parametric Equations

Given the parametric equations, \(x = \sec(\theta)\) and \(y = \tan(\theta)\). These equations represent the x and y coordinates on a graph for a given parameter \(\theta\). Note that the values of \(\sec(\theta)\) become undefined for \(\theta = (2n + 1)\pi/2\) (n is any integer), where the cosine of \(\theta\) equals zero.
02

Convert Parametric Equations to Cartesian Equation

Knowing that \(x = \sec(\theta)\) and \(y = \tan(\theta)\), and also that \(\sec^2(\theta) = 1 + \tan^2(\theta)\), we can convert this to Cartesian coordinates by eliminating the parameter \(\theta\). Thus, substituting \(x\) for \(\sec(\theta)\) and \(y\) for \(\tan(\theta)\), we get: \(x^2 = 1 + y^2\).
03

Graph the Equation

With the Cartesian equation \(x^2 = 1 + y^2\), we visualize the right-half of a circle centered at the origin with radius 1 (not inclusive of zero). Remember that \(x = \sec(\theta)\) will be undefined when x is less than or equal to zero. Thus, we exclude the negative half of the circle.

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