## SOLVING TRIGONOMETRIC EQUATIONS

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

Problem 9.12b:        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. Let's simplify the problem by rewriting it in an equivalent factored form.

The only way the product equals zero is if at least one of the factors equals zero. Therefore, if , or

How do we isolate the x in each of these equations? We could take the inverse (arccosine) of both sides of the equation. 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 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.

We know that Therefore,
If , then

If , then

If , then

If , then

Since the period of equals , these solutions will repeat every units. The exact solutions are

where n is an integer.

The approximate values of these solutions are

where n is an integer.

You can check each solution algebraically by substituting each solution in the original equation. If, after the substitution, the left side of the original equation equals the right side of the original equation, the solution is valid.

You can also check the solutions graphically by graphing the function formed by subtracting the right side of the original equation from the left side of the original equation. The solutions of the original equation are the x-intercepts of this graph.

Algebraic Check:

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 0.84106867 for x, then 0.84106867is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute -0.84106867 for x, then -0.84106867is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 2.30052398 for x, then 2.30052398is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute -2.30052398 for x, then -2.30052398is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 1.04719755 for x, then 1.04719755is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute -1.04719755 for x, then -1.04719755is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 2.0943951 for x, then 2.0943951 is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute -2.0943951 for x, then -2.0943951is a solution.

We have just verified algebraically that the exact solutions are and and these solutions repeat every units.

The approximate values of these solutions are , , and and these solutions repeat every units.

Graphical Check:

Graph the function , formed by subtracting the right side of the original equation from the left side of the original equation. The x-intercepts are the real solutions.

Note that the graph crosses the x-axis many times indicating many solutions. Check the x-intercepts of the graph to see if they match the solutions.

If you would like to review the solution of another problem, click on solution.

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

Author: Nancy Marcus