If you would like an in-depth review of logarithms, the rules of logarithms, logarithmic functions and logarithmic equations, click on logarithmic functions.
Solve for x in the following equation.
Answer: There are an infinite number of solutions: and are the exact solutions, and and x are the approximate solutions.
Solution: To solve for x, first isolate the tangent term.
If we restrict the domain of the sine function to
we can use the arcsine function to solve for x.
The sine function is positive in the first quadrant and the second quadrant. The angle is a reference angle that terminates in the first quadrant. The angle that terminates in the second quadrant that has a reference angle is
The period of sine function is This means that the values will repeat every radians. Therefore, the exact solutions are and where n is an integer. The approximate solutions are and
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=0.37341799
Check the answer x=2.768175
Graph the equation
Note that the graph crosses the x-axis many times indicating many solutions.
Note the graph crosses at 0.37341799 (one of the solutions) and at 2.768175. Since the period of the function is , the graph crosses again at and again at , etc.
If you would like to go back to the equation table of contents, click on contents.
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