## SOLVING TRIGONOMETRIC EQUATIONS

Note:

If you would like an review of trigonometry, click on trigonometry.

Solve for x in the following equation.

Example 1:

There are an infinite number of solutions to this problem. To solve for x, you must first isolate the sine term.

We know that the therefore The sine function is positive in quadrants I and II. The is also equal to Therefore, two of the solutions to the problem are and

The period of the sin function is This means that the values will repeat every radians in both directions. Therefore, the exact solutions are and 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.

Numerical Check:

• Left Side:

• Right Side:        0

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

• Left Side:

• Right Side:        0

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

Graphical Check:

Graph the equation

f (x) = 2 sin(x) - 1

Note that the graph crosses the x-axis many times indicating many solutions. Note that it crosses at . Since the period is , it crosses again at 0.5236+6.283=6.81 and at 0.5236+2(6.283)=13.09, etc. The graph crosses at . Since the period is , it will cross again at 2.618+6.283=8.9011 and at 2.618+2(6.283)=15.18, 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]
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Author: Nancy Marcus