## SOLVING TRIGONOMETRIC EQUATIONS

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, set the equation equal to zero and factor.

then

when when , and when

when and This is impossible because

The exact value solutions are and The approximate value of these 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.

Numerical Check:

Left Side:

Right Side:

Since the left side equals the right side when you substitute 1.5707963for x, then 1.5707963 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute 4.71238898for x, then 4.71238898 is a solution.

Graphical Check:

Graph the equation Note that the graph crosses the x-axis many times indicating many solutions.

Note that it crosses at 1.5707963. Since the period is , it crosses again at 1.5707963+6.2831853=7.85398 and at <tex2htmlcommentmark> 1.5707963+2(6.2831853)=14.137167, etc.

Note that it crosses at 4.71238898. Since the period is , it crosses again at 4.71238898+6.2831853=10.99557 and at <tex2htmlcommentmark> 4.71238898+2(6.2831853)=17.27876, etc.

If you would like to work another example, click on Example.

If you would like to test yourself by working some problems similar to this example, click on Problem.

If you would like to go to the next section, click on next.

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

Author: Nancy Marcus