As Agent Trigonometry, you are given this clue: . If , what is/are the value(s) of .

### Guidance

In the previous concept, we verified trigonometric identities, which are true for every real value of . In this concept, we will solve trigonometric equations. An equation is only true for some values of .

#### Example A

Verify that when .

**
Solution:
**
Substitute in
to see if the equations holds true.

This is a true statement, so is a solution to the equation.

#### Example B

Solve .

**
Solution:
**
To solve this equation, we need to isolate
and then use inverse to find the values of
when the equation is valid. You already did this to find the zeros in the graphing concepts earlier in this chapter.

So, when is the ? Between and . But, the trig functions are periodic, so there are more solutions than just these two. You can write the general solutions as and , where is any integer. You can check your answer graphically by graphing and on the same set of axes. Where the two lines intersect are the solutions.

#### Example C

Solve , where .

**
Solution:
**
In this example, we have an interval where we want to find
. Therefore, at the end of the problem, we will need to add or subtract
, the period of tangent, to find the correct solutions within our interval.

Using the button on your calculator, we get that . Therefore, we have:

This answer is not within our interval. To find the solutions in the interval, add a couple of times until we have found all of the solutions in .

The two solutions are and 4.4806.

**
Concept Problem Revisit
**
To solve this equation, we need to isolate
and then use inverse to find the values of
when the equation is valid.

So now we need to find the values of for which . We know from the special triangles that this value of sine holds true for a angle, but is that the only value of for which it is true?

We are told that . Recall that the sine is positive in both the first and second quadrants, so when also is .

### Guided Practice

1. Determine if is a solution for .

Solve the following trig equations in the interval .

2.

3.

#### Answers

1. Yes, is a solution.

2. Isolate the and then take the square root of both sides. Don’t forget about the !

The at rad (use your graphing calculator). To find the other value where cosine is positive, subtract 0.243 from , rad.

The at rad, which is in the quadrant. To find the other value where cosine is negative (the quadrant), use the reference angle, 0.243, and add it to . rad.

3. Here, we will find the solution within the given range, .

At this point, we can stop. The range of the cosine function is from 1 to -1. is outside of this range, so there is no solution to this equation.

### Practice

Determine if the following values for . are solutions to the equation .

Solve the following trigonometric equations. If no solutions exist, write
*
no solution
*
.

Sole the following trigonometric equations within the interval
. If no solutions exist, write
*
no solution
*
.