SOLVING TRIGONOMETRIC EQUATIONS

Note: If you would like a review of trigonometry, click on trigonometry.

Problem 9.4d:        Solve for x in the following equation.

There are an infinite number of solutions. The exact solutions are and

The approximate solutions are and

Solution:

To solve for x, you must isolate the tangent term. In this problem we do this by rewriting the original equation in an equivalent factored form.

The only way that a product can equal zero is if at least one of the factors equals zero.

The period of tan is , and the period of tan is

Since the period is this means that the rest of the solutions can be found by adding or subtracting multiples of to each of the above solutions. Therefore, the solutions are

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.

Numerical Check:

Left Side:

Right Side:

Since the left side equals the right side when you substitute -0.73006955for x, then -0.73006955 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute 0.392699 for x, then 0.392699 is a solution.

Graphical Check:

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.

Check to see if the graph crosses at the two solutions. It does. Check also to see if the graph crosses to the right and left of each solution by . It does.

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[Algebra] [Trigonometry]
[Geometry] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra]

Author: Nancy Marcus