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Solve for 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 period of is and the period of is The cos x is positive in the firsts and fourth quadrant. This means that the a second solution is
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=2.05824124
Since the left side equals the right side when you substitute 2.05824124for x, then 2.05824124 is a solution.
Check the answer x=6.8429379
Since the left side equals the right side when you substitute 6.8429379for x, then 6.8429379 is a solution.
Graph the equation
Note that the graph crosses the x-axis many times indicating many solutions.
Note the graph crosses at 2.05824124 (one of the solutions). Since the period of the function is , the graph crosses again at 2.05824124+8.901179=10.95942 and again at , etc.
Note the graph crosses at 6.8429379 (one of the solutions). Since the period of the function is , the graph crosses again at 6.8429379+8.901179=15.7441169 and again at , etc.
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