Note: If you would like a review of trigonometry, click on trigonometry.
Problem 9.4a: 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 sine term.
If we restrict the domain of the sine function to
we can use the arcsin function to solve for the
and then x. The reference angle is always
in the first quadrant.
The sine of x is positive in the first and second quadrant and negative in the third quadrant and the fourth quadrant. This means that there are four solutions in the first counterclockwise rotation from 0 to .
One angle, terminates in the first quadrant, a second angle terminates in the second quadrant, a third angle terminates in the third quadrant, and a fourth angle terminates in the fourth quadrant.
Since the period is
this means that the values will repeat every
radians. Therefore, the solutions are
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=0.9553166
Since the left side equals the right side when you substitute 0.9553166for x, then 0.9553166 is a solution.
Check the answer . x=2.186276
Since the left side equals the right side when you substitute 2.186276 for x, then 2.186276 is a solution.
Check the answer . x=4.096909
Since the left side equals the right side when you substitute 4.096909 for x, then 4.096909 is a solution.
Check the answer . x=5.327869
Since the left side equals the right side when you substitute 5.327869 for x, then 5.327869 is a solution.
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 0.9553166, 2.186276, 4.096909, and 5.327869 . Since the period is , the graph will cross the x-axis every units to the left and right of each number.
If you would like to review problem 9.4b, click on problem 9.4b.
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