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Chapter 4: Problem 82

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) in the interval \(-2 \pi, 2 \pi .\) Use the graphs to find real numbers \(x\) such that \(y_{1}=y_{2}\). $$\begin{aligned} &y_{1}=\cos x\\\ &y_{2}=-1 \end{aligned}$$

Short Answer

Expert verified
The solutions to the system of equations \(y_{1}=\cos x\) and \(y_{2}=-1\) in the interval \(-2\pi , 2\pi\) are \(x = -\frac{3\pi}{2}\) and \(x = \frac{3\pi}{2}\).

Step by step solution

01

Graph the functions

Start by plotting both functions \(y_{1}=\cos x\) and \(y_{2}=-1\) within the interval \(-2\pi, 2\pi\). These will be two separate plots on the same graph. The \(\cos x\) plot will look like a wave that waxes and wanes between -1 and 1, while the \(-1\) plot will be a straight horizontal line at y = -1.
02

Identify points of intersection

Next, look for the points where the \(y_{1}=\cos x\) plot intersects the \(y_{2}=-1\) line. These are the points where \(y_{1}\) and \(y_{2}\) are equal.
03

Determine 'x' values

Finally, identify the 'x' coordinates at these points of intersection. These will be the solutions to the given functions within the specified interval. They are the 'x' values for which \(y_{1}\) equals \(y_{2}\).

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