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Solve for the real number x in the following equation.
Answers: There are an infinite number of solutions: and are the exact solutions, and and are the approximate solutions.
To solve for x, first isolate the cosine term.
If we restrict the domain of the cosine function to
we can use the arccos function to
solve for x.
The cosine is negative in the second and third quadrant. The period of this function is . Divide the interval from 0 to into four equal intervals representing quadrants: The cosine is negative in the interval and the solution is The cosine is also negative in the interval and the solution is
Since the period is this means that the values will repeat every radians. Therefore, the exact solutions are and The approximate solutions are and where n is an integer.
These solutions may or may not be the answers to the original problem. You much check them, either numerically or graphically, with the original equation.
Check the answer .x=13.9393
Since the left side equals the right side when you substitute 13.9393 for x, then 13.9393 is a solution.
Check the answer . x=8.0518339
Since the left side equals the right side when you substitute 8.0518339for x, then 8.0518339 is a solution.
Graphical Check: Graph the equation (Formed by subtracting the right side of the original equation from the left side of the original equation.
Note that the graph crosses the x-axis many times indicating many solutions.
Note the graph crosses at 8.05183396 (one of the solutions) as well as 13.9393. Since the period of the function is , the graph crosses again at 8.05183396+21.991149 = 30.04298 and 13.9393 + 21.991149 = 35.930449, etc.
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