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In this video lesson, we are covering the half-angle identities of trigonometry, which are the true statements for half-angles. These are definitions, if you will. In other words, they tell you what a particular trig function equals. You can see here that they take a squared trig function and turn it into a trig function without exponents:
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We have one half-angle for each of our three basic trig functions. We have one for sine, one for cosine, and one for tangent. As you can see, our angle has been halved, hence the name half-angle identity. On the left side, our trig function is squared, and on the right, we see the equivalent statement in terms of cosine without any exponents. Also, the angle on the right side is no longer halved.
So, what can you do with these identities? We use these identities when we need help simplifying a trig function. When we have a squared trig function, it is sometimes difficult to work with in higher math. So, if we turn it into an equivalent statement without exponents, then it will help us solve the problem that much more easily.
You can think of these half-angle trig identities as a key that helps you to decode or simplify a harder problem. Without this key, you might not be able to solve the problem at all. But with the key, you are able to find your way to the answer. Also, these identities are used to prove yet other trig identities or statements. Let's look at a couple of examples now to see what we can do with these trig identities.
This problem is asking us to write this trig function without any exponents:
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At first glance, we might think that this problem is already as simple as it can get. It only has one function: the sine function. But the problem wants us to write it so that we don't have the square. How can we do that?
Well, we look at the angle and notice that our angle is being halved. Ah. We can see if we can use one of our half-angle identities. We look at our list. Oh, look! Here, we have a sine squared function that is equal to something without any exponents. We can use that one to get our answer:
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And there we have our answer without any exponents. This problem was a pretty direct problem with a simple direct answer. Let's look at another problem.
Now, our problem is asking us to prove this trig statement:
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To prove a statement such as this one, we will begin with the left side, since that is the more complicated side. We will leave the right side alone. What we will try to do is to simplify the left side so that it becomes the right side. Let's begin. First, we see that we have a cosine squared function of a half-angle. We can use one of our identities and make that substitution. We get this:
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Okay. Now we can see if we can cancel anything out or otherwise simplify this further. We see that we have a 2 divided by a 2. These cancel each other out. That leaves us with 1 + cos (x) - cos (x). We have a plus cosine and a minus cosine. Don't these add up to 0? They do! So, what are we left with? We are left with 1. That's exactly what we were looking for. Our final answer is our whole string of calculations ending with our desired number of 1:
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Okay, let's review what we've learned now. Half-angle identities are the true statements for half-angles, or definitions, if you will. We have three of them: one for sine, one for cosine, and one for tangent.
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These identities are used to help us prove other trig statements or identities, as well as to help us simplify our more complicated trig problems. We use these identities and substitute them into our problems to get our answers.
When this video lesson ends, take the time to see if you can:
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