SOLVING TRIGONOMETRIC EQUATIONS

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

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

Answers:        There are an infinite number of solutions: and are the exact solutions, and and are the approximate solutions.

Solution:

To solve for x, first isolate the sine term.

If we restrict the domain of the sine function to , we can use the arcsin function to solve for the reference angle , and then x. The reference angle is always in the first quadrant.

The sine of x is positive in the first and second quadrant and negative in the third quadrant and the fourth quadrant. This means that there are four solutions in the first counterclockwise rotation from 0 to .

One angle, terminates in the first quadrant, a second angle terminates in the second quadrant, a third angle terminates in the third quadrant, and a fourth angle terminates in the fourth quadrant.

Since the period is this means that the values will repeat every radians. 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:

Check the answer . x=0.9553166

Left Side:

Right Side:

Since the left side equals the right side when you substitute 0.9553166for x, then 0.9553166 is a solution.

Check the answer . x=2.186276

Left Side:

Right Side:

Since the left side equals the right side when you substitute 2.186276 for x, then 2.186276 is a solution.

Check the answer . x=4.096909

Left Side:

Right Side:

Since the left side equals the right side when you substitute 4.096909 for x, then 4.096909 is a solution.

Check the answer . x=5.327869

Left Side:

Right Side:

Since the left side equals the right side when you substitute 5.327869 for x, then 5.327869 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.

Note the graph crosses at 0.9553166, 2.186276, 4.096909, and 5.327869 . Since the period is , the graph will cross the x-axis every units to the left and right of each number.

If you would like to review problem 9.4b, click on problem 9.4b.

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

Author: Nancy Marcus

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