Solving a Trigonometric Equation Graphically

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  • 0:01 A Trigonometric Equation
  • 0:37 Solving It Graphically
  • 3:17 Finding the Answers
  • 4:02 Things to Watch Out For
  • 4:41 Lesson Summary
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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.

A Trigonometric Equation

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.

Unit Circle
trig graph

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.

Solving It Graphically

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:

Graph for f(x) = cos (x)
trig graph

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.

Lines for x = 0 and x = 2pi
trig graph

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?

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