Note: If you would like a review of trigonometry, click on trigonometry.
Example 1: 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 tangent term.
If we restrict the domain of the tangent function to
we can use the inverse tangent function to solve for reference angle x', and then x.
The reference angle is The tangent function is positive in the first quadrant and in the third quadrant and negative in the second and fourth quadrant.
The period of the
that the values will repeat every
radians in both directions.
Therefore, the exact solutions are
where n is an integer.
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
Right Side: 0
Since the left side equals the right side when you substitute 0.52359877for x, then 0.52359877 is a solution.
Check answer . x=-0.52359877
Right Side: 0
Since the left side equals the right side when you substitute -0.52359877for x, then -0.52359877 is a solution.
Graph the equation Note that the graph crosses the x-axis many times indicating many solutions.
Note that it crosses at 0.52359877. Since the period is , it crosses again at 0.52359877+3.1415927=3.66519 and at <tex2htmlcommentmark> 0.52359877+2(3.1415927)=6.80678, etc.
Note that it crosses at -0.52359877. Since the period is , it crosses again at -0.52359877+3.1415927=2.617899 and at <tex2htmlcommentmark> -0.52359877+2(3.1415927)=5.759587, etc.
If you would like to test yourself by working some problems similar to this example, click on Problem.
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