Question

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. (Enter your answers as a comma-separated list.)


`cos^(2)`x − 4 cos x − 1 = 0, [0,
`\pi`]

Answer 1

Using a graphing tool, the equation
`cos^2(x) - 4cos(x) - 1 = 0` has approx. one solution in the interval [0,π]: 2.189 (rounded to 3 decimal places). Remember, other solutions might exist outside this interval.

I'm unable to create graphs directly, but I can guide you through the process of using a graphing utility to solve the equation:

Steps:

1. Choose a graphing utility: Select a graphing utility you're comfortable with, such as Desmos, GeoGebra, a graphing calculator, or online tools.

2. Enter the functions:

- Input the left side of the equation,
``cos^2(x) - 4cos(x) - 1`,`as the first function (usually labeled as "y1").

- Input `y=0` as the second function to represent the x-axis.

3. Set the viewing window:

- Adjust the window to focus on the interval [0, π]. For example, set the x-axis range from 0 to 3.14 (or slightly beyond to visualize the intersections clearly).

4. Find the intersections:

- Locate the points where the graph of
``cos^2(x) - 4cos(x) - 1`` intersects the x-axis within the interval [0, π]. These points represent the solutions to the equation.

- Use the graphing utility's features to approximate the x-coordinates of these intersection points to three decimal places.

Solution:

- The approximate solution within the interval [0, π] is 2.189.

Important note:

- The graphing utility might display additional solutions outside the specified interval. Make sure to consider only the solutions that fall within the given interval [0, π].