Note: If you would like a review of trigonometry, click on trigonometry.
Example 4: 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 sine term.
If we restrict the domain of the sine function to
we can use the inverse sine function to
solve for reference angle x, and then x. The reference angle is always in
the first quadrant and positive.
The has a period of . This means that it makes one rotation every radians. We can divide the interval into four parts: quadrant I, quadrant II, quadrant III, and quadrant IV. The sine function is positive in the first and second quadrants and negative in the third and and fourth quadrants. In this case, one-fourth of the rotation is 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 period of the
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=0.261799387799
Since the left side equals the right side when you substitute <tex2htmlcommentmark> 0.261799387799 for x, then 0.261799387799 is a solution.
Check answer . x=1.308996939
Since the left side equals the right side when you substitute 1.308996939for x, then 1.308996939 is a solution.
Graph the equation Note that the graph crosses the x-axis many times indicating many solutions.
Note that it crosses two times in the interval from 0 ro and 1.308996939.
Since the period is , the graph crosses again at and etc.
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