Trigonometry: Confirming a Double-Angle Identity

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05/12/2010

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Confirming a Double-Angle Identity
Okay, let's take a look at those sum and differences of trig functions of angle things. So here they are again. We have sine of a sum of two angles equals that; cosine of a sum of two angles equals that; tangent of a sum of two angles equals this. Now let's look at a very special case. Suppose that the two angles, and , were the same. That could happen, right? I never said they were different.
Let's take a look at what happens if I take a look at . So let's have all the be the same. Well, here I would have , . And what would I see here? Here I'd see . So I'd see this term and itself, so I would see . So actually that gives me a formula for the sine of twice an angle in terms of just the angle itself. And that kind of formula is called a double-angle formula. So all I'm going to do is take this formula and replace the and by the same . So I see which is , and that would equal . They're the same thing, so I see . So in fact, if you take a look at this identity, a special case of it would look like this, . And where did that come from? Just this formula here, where I put in the same angle for everybody.
Well, now I can do this for the other trig functions. For example, suppose I do it for cosine. If I put in and I replaced that by , what do I see? I see . That's , . So I'd see a double-angle formula for cosine that would say .
By the way, you may say, "Hey, isn't that equal to 1?" No, no, no. The formula is always equals 1, but their difference in fact doesn't always equal 1. In fact, their difference equals .
Okay. Actually you can do the same trick here with tangent. If I put in I'd see , that's , divided by . That's . So what I see is a formula for a double-angle for tangent, . So these are called double-angle formulas because basically what I'm saying is, if you want to find the trigonometric function of a double-angle--twice some angle--you can do that if you just know the trig functions of the angle itself without the 2 in front.
So I'm going to try to give you an application of this. Suppose you want to find . Well, I don't know that one off the top of my head. These are in radians, by the way; I'm going back to radians. I do know that that's actually . Now is a standard angle. That's actually in layman's terms, but it's radians. So this is actually a double angle. So I can use a double-angle formula for sine and see what it equals just in terms of , something I know.
Now, by the way, should you memorize these formulas? Absolutely not! Since you've already hopefully memorized the sums and differences, all you do is think about that formula with sums but replace by just . So replace all the and by just one and you get this immediately. There's no need to memorize all this stuff. Just think about the other formula and plug in for and .
Okay, so what does this equal? Well, this equals , and those we know because these are all standard angles. So this equals 2 times--well, is and is just . So I can do a little bit of canceling here and I see that this equals . So is actually . See how I can do that? Before it might have been a little bit mysterious. You'd have to use sort of reference angle stuff. Here I can just do it using this formula.
Let's try one last one. Let's try . Well, now I go to this formula here or just think about the actual sum formula for cosine but replacing and just by all , and I would see--well, it's greedy so we first put in . Well, what is ? Well, I already told you that's actually , so squared is , and I subtract. And what's ? Well, I just told you that was . So when I square it I get . So what's ? Well, that equals or also known as . So the cosine of this is . Does that make a little bit of sense? It sure does, because now this angle is actually in the second quadrant, and so the sign in the second quadrant should be positive but the cosine should be negative. So it all plays out just perfectly.
So you can actually use these double-angle formulas in a neat way to actually resolve slightly more exotic angles. We'll take a look at more of these double-angle issues up next. I'll see you there.