*Yuanxin (Amy) Yang Alcocer*Show bio

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*
Show bio

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

A trigonometric identity refers to an equation that is consistently true, regardless of what value is applied to the variable. Learn more about this concept through example equations that are graphed, and discover how to demonstrate whether or not a trigonometric equation is an identity.
Updated: 10/27/2021

In this video lesson, you will learn what to look for in a graph to determine whether a particular equation is a trigonometric identity or not. But first, let's define **trigonometric identity**. A trigonometric identity is any trigonometric equation that is always true for all values of the variable. What does this mean?

This means that if you see an equation and you can say for certain that this equation is always true and you can use the equation with confidence, then this trigonometric equation is a trigonometric identity. Remember identity means the same. Think of the related word 'identical.' You can also think of identity as meaning 'self.' Like when someone asks you, 'What is your identity?' You answer with the kind of person you are or you show your identification card, your identity card.

So, what kinds of equations can you expect to see as we look at trigonometric equations? In math, we already have some trigonometric equations that we know are true. Once we know that a certain trigonometric equation is true, then we can call it a trigonometric identity. For example, these trigonometric equations have been known as trigonometric identities for many, many years:

We use these trig identities to help us solve more complicated trig problems. For example, if we see sine squared plus cosine squared inside a larger problem, we can substitute 1 for that part of the problem, thus making the problem that much simpler and easier to deal with.

Let's take a look at a trigonometric equation that we have been given:

Many times, when we are working with trigonometric equations, you will see a variable such as *x* instead of theta. The letter or symbol that is used to represent our variable doesn't matter. As long as you know which letter is your variable, that is all that matters. So, here *x* is our variable. So, the question now is how can we show that this trigonometric equation is an identity?

We can do that by graphing each side of our equation. So, what we do is we first graph *sin (x) cos (x) tan (x)*, then we graph *1 - cos^2 (x)* to see if they are equal. We can use a graphing calculator to do this or any other graphical method that is easy for you.

Why don't we go ahead and graph each side.

First, the left side, *sin (x) cos (x) tan (x)*. We get this graph:

I've marked three points on this graph so that we can compare these same three points on the next graph to see if they are exactly equal.

Now, let's graph the right side, *1 - cos^2 (x)*:

Hmm. This is interesting. I've marked the same three points on this graph.

So, now we can compare these two graphs to see if our trigonometric equation is an identity. At first glance, they look very identical. So, let's compare our three points. The first point and the last point are the same. What about the middle point? The first graph says (1.571, 1) while the second graph says (pi/2, 1). Are these two points the same point?

Well, what is pi/2? If we go ahead and do the division, we get 1.571.... So, they are the same point. What does this tell us about this trigonometric equation? It is a trigonometric identity. We have proven that our trigonometric equation is a trigonometric identity.

Let's review what we've learned now:

We learned that a **trigonometric identity** is any trigonometric equation that is always true for all values of the variable. In trigonometry, we already have some trigonometric equations that we know are always true and these we call trigonometric identities.

If we are given a trigonometric equation, how do we prove that it is an identity? We can do this graphically by first graphing the left side, then the right side. Then we compare the two graphs to see if they are identical. We can calculate three points for each graph to see if the points are identical. If they are, then our equation is a trigonometric identity.

After reviewing this lesson, you should have the ability to:

- Define trigonometric identity
- Explain how to determine if an equation is a trigonometric identity by graphing it

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