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Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

How can you show someone that you have the right answer? This video lesson will show you how you can do this for a trigonometric equation. Learn what you need to watch for and how to get your answers.

Your best friend since kindergarten comes up to you and begs you to help him solve a **trigonometric equation**, or an equation involving a trig function. The problem is simple enough. It is asking us to find all the solutions between 0 and 2pi for cos (x) = 0. You know how to do this easily by referring to the unit circle.

By looking at the unit circle, you immediately see that the *x* values that give you a 0 value for cos (x) are pi/2 and 3pi/2.

You tell your friend how you easily found the answer, but your friend tells you that he doesn't understand. See, he needs to see how the answer is found. Looking at the unit circle doesn't make sense to him. How does the unit circle translate to the real answers? He needs to see the actual answers. How can you help your friend now?

You can help him by solving the problem graphically. How will this help? This will help your friend because by graphing out the problem, you can show him just how the answers you got relate to the actual graph. You will be able to show him the actual answers.

Because we are going to be graphing the solution, we need to graph the related function to the problem. Our problem is cos (x) = 0. So, what is the function? Since one side of the equation already equals 0, we can simply replace the 0 with f(x). So, our function is f(x) = cos (x). By graphing this function, we will be able to see the solutions where the function equals 0, our problem cos (x) = 0. If we had a problem such as cos (x) = 1, we would simply move the 1 over to the other side by subtracting and then we would replace the 0 with the f(x) to get f(x) = cos (x) - 1.

We can graph out the function, f(x) = cos (x), using various methods. We can use a graphing calculator or we can use a graphing program on the computer. Either way will work; use whichever way is easier for you. I have decided to use a graphing program on the computer. By using this program, I get this kind of graph for f(x) = cos (x). You will get something that looks just like this as well:

I've given you the view of just a short section of the graph. You will see that the graph continues in both directions. The wave is never ending. What we are looking for are the points between 0 and 2pi where the function equals 0 or crosses the *x*-axis. Where are 0 and 2pi? We can graph those lines out too by graphing out *x* = 0 or *x* = 2pi. This helps us mark our area of interest.

Our friend has been watching this whole time and this is making sense to him. So, now what can you tell him about finding the answers?

You can tell him that we are looking for points where the graph, our line, crosses the *x*-axis. Why? Because at those points, our function equals 0 and we have cos (x) = 0. And so, those points answer our original problem. You show him that there are two such points that satisfy our problem of cos (x) = 0. You point them out to him. Now you use your graphing calculator or program to help you find what those points are.

Your friend has an 'Aha!' moment. Now he sees the answers. They are, as you told him earlier, pi/2 and 3pi/2. It all makes sense to him now.

Before we finish up, I want to share with you some things that you need to be watchful of as you solve equations graphically.

First, be watchful of the domain that is given to you. Remember that your domain is the range of possible inputs or possible solutions. It gives you the area of interest on your *x*-axis.

Second, there may be more than one answer. Just because you have found one, doesn't mean you are done. Sometimes there are two or even more answers. And sometimes, you can have none. It all depends on your function and the domain that is given to you.

So, now let's review:

A **trigonometric equation** is an equation involving a trig function. We can solve it graphically by graphing the related function. For example, the trigonometric equation cos (x) = 1 has the related function f(x) = cos (x) - 1, found by moving all the terms to one side and replacing the 0 on the one side with f(x). We can graph our function using either a graphing calculator or a graphing program. Once we have graphed it, we mark our area of interest based on the domain that is given and then we look for the answers where the graph crosses the *x*-axis, where our function equals 0. Depending on the function and the domain, we may find no answers, one answer, two answers, or even more answers.

After you've reviewed this video lesson, you should be able to:

- Define trigonometric equation
- Explain how to solve a trigonometric equation graphically
- Identify precautions to take when solving trigonometric equations graphically

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Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

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