SOLVING TRIGONOMETRIC EQUATIONS

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

Problem 9.5a: Solve for x in the equation

where n is an integer.

The approximate values of these solutions are

where n is an integer.

Solution:

There are an infinite number of solutions to this problem. To solve for x, set the equation equal to zero and factor.

then

The function when and when x=0 or .

when and

How do we isolate the x in the equations ? We could take the arccosine of both sides. However, the cosine function is not a one-to-one function.

Let's restrict the domain so the function is one-to-one on the restricted domain while preserving the original range. The graph of the cosine function is one-to-one on the interval If we restrict the domain of the cosine function to that interval , we can take the arccosine of both sides of each equation.

Since we know that if then and

The exact solutions are

where n is an integer.

The approximate values of these 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 for x, then 0 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute 3.141592653 for x, then 3.141592653 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute 0.785398 for x, then 0.785398 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute 2.35619447for x, then 2.35619447 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute -0.785398for x, then -0.785398 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute -2.35619447 for x, then -2.35619447 is a solution.

Graphical Check:

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

Verify that it crosses at 0. Since the period is , it crosses again at 0+6.2831853=6.2831853 and at 0+2(6.2831853)=12.5663706, etc.

Verify that it crosses at 3.141592653. Since the period is , it crosses again at 3.141592653+6.2831853=9.42477796 and at 3.141592653+2(6.2831853)=15.7079633, etc.

Verify that it crosses at 0.785398. Since the period is , it crosses again at 0.785398+6.2831853=7.0685833 and at 0.785398+2(6.2831853)=13.3517686, etc.

Verify that it crosses at 2.35619447. Since the period is , it crosses again at 2.35619447+6.2831853 = 78.639379777 and at 2.35619447+2(6.2831853)=14.922565, etc.

Verify that it crosses at -0.785398. Since the period is , it crosses again at -0.785398+6.2831853=5.4977873 and at -0.785398+2(6.2831853)=11.7809726, etc.

Verify that it crosses at -2.35619447. Since the period is , it crosses again at -2.35619447+6.2831853=3.9269908 and at -2.35619447+2(6.2831853)=10.210176, etc.

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

Author: Nancy Marcus