Note: If you would like a review of trigonometry, click on trigonometry.
Example 2: Solve for x in the following equation.
There are an infinite number of solutions to this problem. To solve for x, you must first isolate the cosine term.
If we restrict the domain of the cosine function to
we can use the inverse cosine function to solve for reference
and then x. The reference angle is always located in
the first quadrant and positive.
The cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants. We will use the reference angle to find the four angles.
The first solution is the angle that terminates in the first quadrant: is The second solution is the angle that terminates in the second quadrant: . The third solution is the angle that terminates in the third quadrant: The fourth solutions is the angle that terminates in the fourth quadrant:
The period of the cos
This means that
the values will repeat every
radians in both directions. Therefore,
the exact solutions are
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 answer . x=0.52359877
Since the left side equals the right side when you substitute 0.52359877for x, then 0.52359877 is a solution.
Check answer . x=2.6179938
Since the left side equals the right side when you substitute 2.6179938for x, then 2.6179938 is a solution.
Check answer .x=3.66519
Since the left side equals the right side when you substitute 3.66519 for x, then 3.66519 is a solution.
Check answer . x=5.759586537
Since the left side equals the right side when you substitute 5.759586537for x, then 5.759586537 is a solution.
Graph the equation Note that the graph crosses the x-axis many times indicating many solutions.
Note that it crosses four time in the interval from 0 ro , 3.66519 and 5.759586537.
Since the period is , the graph crosses again at , , and etc.
If you would like to test yourself by working some problems similar to this example, click on Problem.
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