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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 sine term.
If we restrict the domain of the cosine function to
we can use the arcsin function to solve for x.
The sine of x is negative in the third quadrant and the fourth quadrant. This means that there are two solutions in the first counterclockwise rotation from 0 to .
One angle, terminates in the third quadrant and angle terminates in the fourth quadrant.
Since the period is this means that the values will repeat every radians. Therefore, the 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=3.58450369766
Since the left side equals the right side when you substitute 3.58450369766for x, then 3.58450369766 is a solution.
Check the answer . x=5.840274
Since the left side equals the right side when you substitute 5.840274 for x, then 5.840274 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 3.584503697667 ( one of the solutions ). Since the period of the function is , the graph crosses again at 3.584503697667+6.2831853=9.867689 and again at , etc.
The graph also crosses at 5.840274 ( another solution we found ). Since the period is , it will cross again at 5.840274+6.2831853=12.123459 and at , etc
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