# Precalculus : Solving Trigonometric Equations and Inequalities

## Example Questions

### Example Question #1 : Solving Trigonometric Equations And Inequalities

Use trigonometric identities to solve the following equation for :

Explanation:

Use the trigonometric identities to switch sec into terms of tan:

hence,

So we have , making

Therefore the solution is  for n being any integer.

### Example Question #1 : Solving Trigonometric Equations And Inequalities

Which of the following is not a solution to   for

Explanation:

We begin by setting the right side of the equation equal to 0.

The equation might be easier to factor using the following substitution.

This gives the following

This can be factored as follows

Therefore

Replacing our substitution therefore gives

Within our designated domain, we get three answers between our two equations.

when

when

Therefore, the only choice that isn't correct is

### Example Question #3 : Solving Trigonometric Equations And Inequalities

Find one possible value of .

Explanation:

Begin by isolating the tangent side of the equation:

Next, take the inverse tangent of both sides:

Divide by five to get the final answer:

### Example Question #4 : Solving Trigonometric Equations And Inequalities

Use trigonometric identities to solve for the angle value.

Explanation:

There are two ways to solve this problem. The first involves two trigonometric identities:

The second method allows us to only use the first trigonometric identity:

### Example Question #5 : Solving Trigonometric Equations And Inequalities

Use trigonometric identities to solve the equation for the angle value.

Explanation:

The simplest way to solve this problem is using the double angle identity for cosine.

Substituting this value into the original equation gives us:

### Example Question #6 : Solving Trigonometric Equations And Inequalities

According to the trigonometric identities,

Explanation:

The trigonometric identity , is an important identity to memorize.

Some other identities that are important to know are:

### Example Question #7 : Solving Trigonometric Equations And Inequalities

If  exists in the domain from  , solve the following:

Explanation:

Factorize .

Set both terms equal to zero and solve.

This value is not within the  domain.

This is the only correct value in the  domain.

### Example Question #8 : Solving Trigonometric Equations And Inequalities

Solve for  in the equation  on the interval .

Explanation:

If you substitute  you obtain a recognizable quadratic equation which can be solved for

.

Then we can plug  back into our equation and use the unit circle to find that

.

### Example Question #9 : Solving Trigonometric Equations And Inequalities

Given that theta exists from , solve:

Explanation:

In order to solve  appropriately, do not divide  on both sides.  The effect will eliminate one of the roots of this trig function.

Substract  from both sides.

Factor the left side of the equation.

Set each term equal to zero, and solve for theta with the restriction .

Solve  for