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Chapter 5: Problem 77

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. $$4 \cos ^{2} x-2 \sin x+1=0, \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

Short Answer

Expert verified
Since the solutions need a graphical approach, the precise answers may depend on the graphing tool used and the accuracy of the approximation tools it provides. Thus, definitive solutions cannot be given, but they will be within the interval \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) and can be approximated to three decimal places using a graphing tool.

Step by step solution

01

Understand the equation

The given equation is a mix of sine and cosine functions: \(4 \cos ^{2} x-2 \sin x+1=0\). The goal is to find the points \(x\) within the interval \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) where this equation equals zero.
02

Plot the equation using a graphing utility

Input the equation into a graphing utility, ensuring the mode is set to radians because the interval is given in terms of \(\pi\). The graph should display the function within the specified interval \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
03

Identify the x-intercepts

The values of \(x\) at which the equation equals zero correspond to points where the graph crosses the x-axis (x-intercepts). Verify that these x-intercepts fall within the given interval.
04

Approximate the solutions

Use the graphing utility's tool for finding x-intercepts (some tools may refer to this as 'roots' or 'zeroes') to get the decimal approximations to three decimal places. Ensure to verify if the solutions are within the valid range \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\)

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