Answered: y y = - sin 1+ -4 4π

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Use the graph (see given herewith) to obtain the graph of the corresponding reciprocal function, cosecant. Give the equation of the function for the graph that you obtain.

Expert Solution

Step 1: Given

The given graph of the function $y = - \frac{1}{2} \sin (\frac{x}{2})$ .

We need to sketch the graph of the reciprocal function and find its equation.

Step 2: Graph of the reciprocal function

Let $y = g (x)$ be the reciprocal function.

From the given graph,

As $x$ approaches to $- 2 π$ from the left, the graph value is negative and approaches to $0$ .

Thus $\lim_{x \to - 2 π^{-}} g (x) = - \infty$ .

As $x$ approaches to $- 2 π$ from the right, the graph value is positive and approaches to $0$ .

Thus $\lim_{x \to - 2 π^{+}} g (x) = \infty$ .

As $x$ approaches to $0$ from the left, the graph value is positive and approaches to $0$ .

Thus $\lim_{x \to 0^{-}} g (x) = \infty$ .

As $x$ approaches to $0$ from the right, the graph value is negative and approaches to $0$ .

Thus $\lim_{x \to 0^{+}} g (x) = - \infty$ .

Using these limits and the graph of the cosine function.

The graph of the reciprocal function of $y = - \frac{1}{2} \sin (\frac{x}{2})$ as follows:

The equation of the reciprocal function,

$y = \frac{1}{- \frac{1}{2} \sin (\frac{x}{2})} y = - 2 c s c (\frac{x}{2}) .$

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