Answer:
x ∈ {60°, 120°, 240°, 300°}
Explanation:
a) The graph is shown below. It shows solutions to be ...
x ∈ {60°, 120°, 240°, 300°}
The graph shows the solutions to f(x) = 0, where we defined f(x) to be the difference between the two sides of the equation:
f(x) = 4cos(x)^2 -2cos(x) -(1 -2cos(x))
We chose this method because this graphing calculator easily highlights x-intercepts. Those are the solutions to the given equation.
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b) The second attachment shows a "numerical" solution. We assume you're interested in a solution in which x-values are tried until suitable solution values are found. The attached spreadsheet has columns for various values of x, the left side of the equation, the right side of the equation, and the difference between the two. Conditional formatting has been used to highlight differences of zero. These correspond to values of x that satisfy the equation. The solutions are ...
x ∈ {60°, 120°, 240°, 300°}
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c) We can add 2cos(x) to both sides of the equation, then divide by 4 to get a quadratic in cos(x). Taking the square root gives the values of cos(x) corresponding to angles in all four quadrants. When these are suitably translated to the angle range of interest, we have our solution.
cos(x)^2 = 1/4
Taking the square root gives two values of cos(x):
cos(x) = ±√(1/4) = ±1/2
The corresponding values of x are ...
x = ±60° +k·180° . . . . . for any integer k.
The x-values in the range of interest are ...
x ∈ {60°, 120°, 240°, 300°}