WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi, welcome back to the trigonometry lectures on educator.com.
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Today, we're going to learn about the double angle formulas, so here they are.
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The first one is sin(2x)=2sin(x)×cos(x).
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You may think there's so many formulas to remember in trigonometry.
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This one, if you have trouble remembering it, you can work it out from the addition formula.
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You do have to remember something, but if you can remember the sin(a+b)=sin(a)×cos(b)+cos(a)×sin(b).
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If you remember that one, then you don't really need to learn anything new here because you can work it out so quickly.
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Just take a and b, both to be x in the addition formula.
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If a is x and b is x, then what you get here is sin(2x)=sin(x)×cos(x)+cos(x)×sin(x).
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What you get is just 2sin(x)×cos(x).
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If you can remember the addition formula, the double angle formulas are really nothing new to remember here, same goes for the cos(2x) formula.
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If you remember the addition formula for cosine, you might want to try just plugging in x for each of the a's and b's, and you'll see that what you get is exactly cos²(x)-sin²(x).
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Now, there's two other ways that you often see this formula written as 2cos²(x)-1, and 1-2sin²(x).
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Those might look different but actually you can figure them out very quickly, or check them very quickly, because 2cos²(x)-1 is 2cos²(x) minus, now remember 1 is the same as sin²(x)+cos²(x).
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If you work with that a little bit, you have 2cos²-cos².
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That's just a single cos²(x)-sin²(x), and so all of a sudden this goes back to the original formula for cos(2x).
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You could do this, you can check the second formula the exact same way, if you convert the 1 into sin²+cos², you'll see that it converts back into this original formula for cos(x).
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Even though it looks like there's 4 new formulas to remember here, really the basic sin(2x) and cos(2x), you can work both of those out from the additional formulas.
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The other two formulas for cosine, you can just work them out if you remember the original formula for cosine and then the Pythagorean identity, sin²+cos²=1, which certainly any trigonometry student is going to remember the Pythagorean identity.
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It's really not a lot of new memorization for these formulas.
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The more interesting question here is how are you going to use them.
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Let's try them out on some examples.
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Our first example here is we're just going to get some practice using the sine and cosine of 2x formulas, the double angle formulas.
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To find the sine and cosine of 2π/3.
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Even though 2π/3 is a common value, hopefully you can work out the sine and cosine of 2π/3 without using the double angle formulas.
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We're going to try them out using the double angle formulas, and then we'll just check that the answers we get agree with the values that we know coming from the common values.
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We'll use that as a check, we won't use that at the beginning.
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We're also going to use all three of the formulas for cosine and just check and make sure that they all work out, that they all agree with each other.
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Let's start out by remembering those, actually, four formulas, sin(2x) is 2sin(x)×cos(x), and cos(2x) is cos²(x)-sin²(x).
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Here, we're being asked to find the sine and cosine of 2π/3.
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We're going to use x=π/3, that way 2x is 2π/3.
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So, sin(2π/3), using x=π/3, it's 2sin(π/3)×cos(π/3).
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I remember that the sin(π/3), that's a common value, so the sin(π/3) is root 3 over 2, cos(π/3) is 1/2, the 2 and that in 1/2 cancel, and what we'll get is root 3 over 2.
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Now, let's try the cosine, cos(2π/3), is cos²(π/3)-sin²(π/3) according to our formula, but we're going to check it out and see if it works.
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Now, cos(π/3) is 1/2, so (1/2)² minus the sin(π/3) is root 3 over 2, we'll square that out.
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1/2 squared is 1/4, root 3 over 2 squared is, root 3 squared is 3, 2 squared is 4, we get 1/4-3/4=-1/2.
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Now, there were two other formulas for cos(2x), we want to check out each one of those, cos(2x)=2cos²(x)-1.
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It was also supposed to be equal to 1-2sin²(x).
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We're going to check out each one of those.
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Cos(2π/3), using those other formulas, is equal to 2cos²(π/3)-1, which is 2.
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Now, cos(π/3), that's a common value, that's 1/2, (1/2)²-1, which is 2×1/4-1, which is 1/2-1, is -1/2.
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Let's use the other version, 1-2sin²(π/3), we'll use the last cosine formula there.
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That's 1-2, now, sin(π/3), I remember that's a common value, root 3 over 2.
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We're going to square that out, that's 1-2 times, root 3 squared is 3, and 2² is a 4.
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That's 1-3/2=-1/2.
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The first thing we noticed is that these 3 different formulas for cos(2x) they all gave us the answer -1/2.
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They do check with each other, that's reassuring.
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Now, let's work out the sine and cosine of 2π/3 just using the old-fashioned common values.
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Let me draw my unit circle.
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There's 0, π/2, π, and 3π/2.
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2π/3 is 2/3 the way from 0 to π.
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There it is right there.
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That's my 30-60-90 triangle, so I know the values there are root 3 over 2 and 1/2.
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I just have to figure out the sine and cosine, which ones are positive and which ones are negative.
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I know that the cos(2π/3) because that's the x-value, and the x-value is negative, that's -1/2.
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The sin(2π/3) is the y-value, which is positive, that's root 3 over 2.
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We worked those out just looking at the unit circle and remembering the common values but that checks out with the values we got from the formulas there sin(2π/3) and each one of the formulas for cos(π/3).
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What we're doing there is working out each one of the formulas for sin(2x) and cos(2x) with x=π/3.
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That separates it out into expressions in terms of sines and cosines of π/3, which I remember so I just plug those in and I get the sine and cosine of 2π/3.
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All the cosine formulas agree with each other and they all check with the values that I can find just by looking at the unit circle.
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Our next example is to use the double angle formulas to prove a trigonometric identity.
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It's not so obvious how to start with this one.
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We're actually going to start with the right-hand side because it looks more complicated.
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I'm going to start with the right-hand side and that's 2tan(x)/1+tan²(x).
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I'm evaluating the right-hand side, I'm going to work with it a bit and hopefully I can simplify it down to the left-hand side, but we'll see how it goes.
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First thing, I'm going to do is to change everything into sines and cosines.
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That's a good rule when you're not sure what to do with the trigonometric identity is to change everything into sines and cosines.
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If you got a tangent or a secant, or a cosecant or a cotangent, convert it into sines or cosines.
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It will probably make your life easier.
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I'll write this as 2, tangent, remember is sin/cos, and 1+tan², that's 1+sin²(x)/cos²(x).
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Now, I see a lot of cosines in denominators here, I think we're going to try to clear those out.
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We multiply top and bottom by cos²(x) and see what happens with that.
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That's multiplying by 1, so that's safe.
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On the top, I have 2sin(x), now I had a cos(x) in the denominator but I multiplied by cos², that gives me cos(x) in the numerator.
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In the bottom, I have cos² times 1+sin²(x) over cos², that gives me 1×cos² is cos²(x), plus the cos²(x) cancels with the denominator sin²(x).
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Now, look at this, the top is exactly 2sin(x)×cos(x).
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I remember that, that's my formula for sin(2x).
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Now the bottom, that's the Pythagorean identity, so that's just 1, cos²+sin²(x) is 1.
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This converts into sin(2x), but that's equal to the left-hand side of what we were trying to prove.
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We started with the right-hand side because it looked a little more complicated there.
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I see a bunch of tangents, I am not so sure what to do with those, I convert them into sines and cosines.
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I see a lot of cosines in the denominator, so I multiply top and bottom by cos²(x).
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Then I start noticing some formulas that I recognized, 2sin(x)×cos(x) is a double angle formula, and cos²(x)+sin²(x), that's the Pythagorean identity.
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It reduces down into the right-hand side.
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Let's try another example here, we're going to use the addition and subtraction formulas to derive a formula for tan(2x) in terms of tan(x).
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Remember, we have formulas for sin(2x) and cos(2x), we're going to find a formula for tan(2x) just in terms of tan(x).
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When we get that, we're going to check the formula on a common value π/6, because I know what the tangent of that is, and I know what the tan(2x) is, so we can check whether our formula works.
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Let me start out with, tan(2x), don't know much about that except that the definition of tan(2x) is sin(2x)/cos(2x).
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Now, I'm going to use, well, it's the addition and subtraction formulas but it's really the double angle formulas.
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Of course, those come from the addition and subtraction formulas.
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Now, sin(2x) is 2sin(x)×cos(x), that's the double angle formula for sine.
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Of course, you find that out from the addition formula.
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Cos(2x) is cos²(x)-sin²(x), that was the first double angle formula for cosine.
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Now, it's not totally obvious how to proceed next, but I know that I'm trying to get everything in terms of tan(x).
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Right now, I've got a bunch of cosines lying around, I'd like to move those down into the denominator.
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The reason is because tangent is sin/cos, so I would like to be dividing by cosines.
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What I'm going to do is I'm going to divide the top by cos²(x), and I'll divide the bottom by cos²(x).
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We're dividing top and bottom by cos²(x), that's dividing by 1, so that's legitimate, we'll see what happens.
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Now, in the numerator, we get 2sin(x), we had a cos(x) before, we divided by cos², we get 2sin(x)/cos(x).
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In the bottom, we're dividing everything by cos²(x), we get 1-sin²(x)/cos²(x).
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That's really nice because now we have sin/cos everywhere and that's tangent.
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We are asked to find everything in terms of tan(x).
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What we get here is 2sin/cos is tan(x) over 1-sin²(x)/cos²(x) is tan²(x).
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Our formula, our double angle formula for tangent is tan(2x)=2tan(x)/(1-tan²(x)).
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Now, I didn't list this at the beginning of the lecture as one of the main formulas that you really need to memorize.
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It kind of depends on your trigonometry class, in some classes they will ask you to memorize this formula, this formula for tan(2x).
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I don't think it's worth memorizing.
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In my trigonometry classes, I don't require my students to memorize these formulas for tan(2x).
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I do require them to memorize sin(2x) and cos(2x) and I figure they can work out the other ones from that.
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You may have a teacher who requires you to memorize the formula for tan(2x).
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If so, here it is, here is the formula that you want to remember.
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Let's check that out on a value that I already know the tangent of, let's try x=π/6.
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The tan(2π/6), according to this formula, would be 2×tan(π/6)/(1-tan²(π/6)).
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Now, π/6 is a common value, tan(π/6), I remember that, I've got that one memorized, it's root 3 over 3.
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If you don't have that one memorized, it probably is a good one to memorize, but if you don't have it memorized, you can work it out as long as you remember sine and cosine of π/6.
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You just divide them together and get the tan(π/6).
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This is 2 times root 3 over 3, over 1 minus root 3 over 3 squared.
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Let's do a little over that, that's 2 times root 3 over 3, over 1 minus root 3 over 3 squared, is 3, over 3 squared is 9.
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That's 3/9 which is 1/3.
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This is 2 root 3 over 3, divided by 2/3.
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Remember how you divide fractions, you flip it and multiply, 3/2, that cancels off the 2 and the 3, this whole thing boils down to just a root 3 as tan(2π/6).
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Of course, 2π/6 is just π/3.
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π/3 is another common value that I know the tangent of.
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tan(π/3), I remember, is root 3, that's a common value.
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Again, if you don't remember that, remember the sine and cosine of π/3, divide them together and you'll get root 3.
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Look at that, our answers agree.
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That confirms our formula for tan(2x).
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To recap the important parts of that problem, we have to figure out tan(2x).
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We wrote it as sin/cos of 2x.
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We expanded each one of those using the double angle formulas that we learned at the beginning of the lesson.
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Then, I was trying to get this in terms of tan(2x).
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I wanted to get some cosines in the denominator, that's why I divided top and bottom by cos²(x).
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That converted the thing into something in terms of tan(x).
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Then we checked that out by plugging in x=π/6, that's something that I know the tangent of, worked through the formula, and we got an answer square root of 3.
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That checks the common value that I also know tan(π/3) is square root of 3.
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We'll try some more examples of that later.