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

**Problem** 9.7c: **Solve for x in the equation **

Answer: The exact answers are

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.

In the equation
there are two different
trigonometric terms. To simplify the solution, convert the equation to an
equivalent equation with just sine terms of just cosine terms. Which one
shall it be? The easiest way is to convert the trigonometric terms to sine
terms because you can solve for
in the identity

Substitute for in the original equation.

Rewrite the equation in factored form and solve for .

The only way the product of two factors equals zero is if at least one of the factors equals zeros. This means that if or

We have transformed a difficult problem into two problems. To find the solutions to the original equation, , we first solve for in the equations and

and

To solve for x, we must isolate the x. How do we do that? We could take the <*tex*2*html*_{c}*omment*_{m}*ark*>
*arcsine* of both sides. However, the *sine* function is not a one-to-one
function.

We can restrict the domain of the
function so it is one-to-one on
the restricted domain while preserving the original range. The *sine*function is one-to-one on the interval
If we restrict the domain of the *sine* function to that
interval , we can take the *arcsine* of both sides of each equation.

We know that Therefore, 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 the left side of the original equation and graphing the function formed
by the right side of the original equation. The x-coordinates of the points
of intersection are the solutions. The right side of the equation is 0 and <*tex*2*html*_{c}*omment*_{m}*ark*>
*f*(*x*)=0 is the x-axis. So really what you are looking for are the
x-intercepts to the function formed by the left side of the equation.

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
0.52359878 for x, then
0.52359878is a solution.**
**

**
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**
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**
**

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 2.6179939 for x, then 2.6179939 is
a solution.**
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**

**
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**
**

The exact solutions are and and these solutions repeat every units. The approximate values of these solutions are and 2.6179939 and these solutions repeat every units.

Graphical Check:**
**

**
**

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

Verify the graph crosses the x-axis at
0.52359878. Since the period is
,
the graph also crosses the x-axis again at
<*tex*2*html*_{c}*omment*_{m}*ark*>
0.52359878+6.2831853=6.806784 and at
,
etc.

Verify the graph crosses the x-axis at
.
Since the period is
,
the graph also crosses the x-axis again at
<*tex*2*html*_{c}*omment*_{m}*ark*>
2.6179939+6.2831853=8.901179 and at
,
etc.

Note: If the problem were to find the solutions in the interval , then you choose those solutions from the set of infinite solutions that belong to the set and

If you would like to review the solution to another problem, click Back to Solutions and then click on the word solution opposite the problem you want to review.

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