Math Analysis Double Angle Formulas

Hi we are trying some examples of the double angle formulas for sin and cos.0000

We are going to try to find the sin and cos of 4pi/3.0005

We are going to use all three cos formulas and check that they agree.0010

We are also going to use our common values to find the sin and cos of 4pi/3 to check our answers.0014

Let me write down the double angle formulas that we are going to be using.0021

We are going to use sin 2X = 2 sin X cos X.0025

These are probably worth remembering, but if you do not remember them, you can work them out from the addition formula for sin and cos.0034

Cos(2X) is cos x ² - sin x ².0043

Of course, the X here would have to be 2pi/3 because what we are really trying to find is the sin and cos of 4pi/3.0051

So, sin(4pi/3) according to our double angle formula is equal to 2 x sin(2pi/3) x cos(2pi/3).0061

Now, 2pi/3 that is a common value, I remember its sin and cos, its sin is root 3/2 and its cos is -1/2.0078

It is negative because 2pi/3 is in the second quadrant, so its X coordinate is negative.0089

Remember, cos is the X coordinate and this simplifies down to cancel and we will get –root 3/2.0094

cos(4pi/3) is cos(2pi/3)² - sin(2pi/3)².0106

So, plug those common values in the cos(2pi/3) is -1/2 and sin(2pi/3) is positive root 3/2.0122

I will get ¼ negative goes away because they got squared – root 3 squared is ¾ and so I get -1/2.0137

That was the first of the three formulas for cos(2x).0151

Let me remind you what the other two formulas are.0154

cos(2x) is equal to 2 cos X ² - 1 and the other version we have of that formula is 1 – 2 sin X².0156

These are all different formulas for cos(2x) and we will try each one now.0177

The first one there is cos(4pi/3) is equal to 2 cos(2pi/3) ² - 1.0183

The cos(2pi/3) is -1/2 because it is in the second quadrant -1, that is (2 x ¼ - 1), which is ½ - 1, which is – ½.0201

If we use the other version of the formula we will get cos(2pi/3) is (1- 2 sin(2pi/3)²) which is (1-2 x (root 3/2)²).0225

2pi/3 is our common value, I remember its sin, (1 – 2 x root 3 squared is 3), 4 in the denominator so 1 – 3 (1/2), and again we will get -1/2.0243

That is very reassuring because if you look at the three different formulas for cos, we got the same answer for all three of them.0262

That was the first point we wanted to check, but now let us check using our common values, there is 0, pi/2, 3pi/2, 2pi.0272

Now, 4pi/3 is bigger than pi, it is down here.0294

4pi/3 is 2/3 around the unit circle to 2pi.0300

That is one of our common triangles, that is the 30, 60, 90 triangles.0306

I know that the length of the sides there are root 3/2 and 1/2, I can figure out the sin and cos from that.0311

I just have to figure what are the positive or negative.0320

Well, the cos(4pi/3) is negative because the X value is negative, so it is – ½.0323

The sin(4pi/3) is also negative because the Y value is also negative there, it is – root 3/2.0334

Those are the answer we get using the common values on the unit answer.0346

But if you look, that is also the answers we got using the double angle formula breaking 4pi/3 up into ((2 x (2pi/3)).0351

We got sin, was – root 3/2, and cos was – ½.0363

It indeed, in fact, agrees with the values that we got from the unit circle.0367

Finally we are going to use the double angle formulas to prove another trigonometric identity.0000

We are going to prove that sec(2x) is equal to (sec X ²) / (2 – sec X ²).0005

We are going to start with the right side because it looks more complicated.0013

The right hand side and we will try to manipulate it to the left hand side.0017

Let me start with the right hand side, (sec X²) / (2 – sec X ²).0022

Again, we do not know what to do with the trigonometric identity, it is often good to start with the more complicated side.0035

Secondly, convert everything to sin and cos.0042

Here I got a lot of sec, I am going to convert it to the definition of sec is (1/cos), this is (1/cos X ²).0046

My denominator, I have 2 – 1/ (cos X ²).0055

I see a lot of cos ² in the denominator.0064

I think I’m going to try to clear that by multiplying top and bottom by cos X ².0068

On the top, I will just get 1, on the bottom I get 2 cos X ² - 1.0077

But look at that, 2 cos X ² - 1.0090

That is one of the formulas that I remember for cos(2x), this is 1/cos(2x).0094

Now, let us remember by definition, one of our cos is exactly the same as sec.0104

So, this is sec(2x) and that is the left hand side of the identity that we are trying to prove.0110

We proved that we started with the right hand side, we derived the left hand side.0120

The key things to notice in there, the way it worked was, first of all the right hand side is a little more complicated, so we are going to work on that one.0124

When I see a bunch of sec, I try to convert it into sin and cos because I know how to manipulate sin and cos.0131

I got more formulas for them than for sec, tan, cosec, and cot.0138

I converted into sin and cos.0144

I see some cos in the denominator, I decided to multiply by cos X ² to clear away those denominators.0147

I'm multiplying that thru and there is really some pattern recognition here knowing your identity formulas.0155

When I see that 2 cos X ² - 1, a little bell goes of in my head, “wait I have seen that somewhere before, oh yes that is equal to cos(2x)”.0162

Now I got 1/cos(2x), that is by definition sec(2x) and so I converted into the left hand side.0174

That is how you can use the double angle identities to prove more complicated trigonometric identities.0184

That is the end of our lecture on double angle identities.0192

These are the trigonometry lectures for www.educator.com.0196

Hi, welcome back to the trigonometry lectures on educator.com.0000

Today, we're going to learn about the double angle formulas, so here they are.0004

The first one is sin(2x)=2sin(x)×cos(x).0008

You may think there's so many formulas to remember in trigonometry.0014

This one, if you have trouble remembering it, you can work it out from the addition formula.0019

You do have to remember something, but if you can remember the sin(a+b)=sin(a)×cos(b)+cos(a)×sin(b).0024

If you remember that one, then you don't really need to learn anything new here because you can work it out so quickly.0041

Just take a and b, both to be x in the addition formula.0046

If a is x and b is x, then what you get here is sin(2x)=sin(x)×cos(x)+cos(x)×sin(x).0052

What you get is just 2sin(x)×cos(x).0068

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.0074

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).0082

Now, there's two other ways that you often see this formula written as 2cos²(x)-1, and 1-2sin²(x).0096

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).0106

If you work with that a little bit, you have 2cos²-cos².0126

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).0132

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).0143

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.0158

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.0172

It's really not a lot of new memorization for these formulas.0185

The more interesting question here is how are you going to use them.0190

Let's try them out on some examples.0194

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.0197

To find the sine and cosine of 2π/3.0206

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.0211

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.0219

We'll use that as a check, we won't use that at the beginning.0232

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.0235

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).0243

Here, we're being asked to find the sine and cosine of 2π/3.0264

We're going to use x=π/3, that way 2x is 2π/3.0267

So, sin(2π/3), using x=π/3, it's 2sin(π/3)×cos(π/3).0277

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 2.0294

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.0314

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.0333

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=-1/2.0344

Now, there were two other formulas for cos(2x), we want to check out each one of those, cos(2x)=2cos²(x)-1.0357

It was also supposed to be equal to 1-2sin²(x).0370

We're going to check out each one of those.0376

Cos(2π/3), using those other formulas, is equal to 2cos²(π/3)-1, which is 2.0379

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 -1/2.0393

Let's use the other version, 1-2sin²(π/3), we'll use the last cosine formula there.0413

That's 1-2, now, sin(π/3), I remember that's a common value, root 3 over 2.0425

We're going to square that out, that's 1-2 times, root 3 squared is 3, and 2² is a 4.0433

That's 1-3/2=-1/2.0443

The first thing we noticed is that these 3 different formulas for cos(2x) they all gave us the answer -1/2.0452

They do check with each other, that's reassuring.0460

Now, let's work out the sine and cosine of 2π/3 just using the old-fashioned common values.0463

Let me draw my unit circle.0471

There's 0, π/2, π, and 3π/2.0482

2π/3 is 2/3 the way from 0 to π.0490

There it is right there.0493

That's my 30-60-90 triangle, so I know the values there are root 3 over 2 and 1/2.0496

I just have to figure out the sine and cosine, which ones are positive and which ones are negative.0505

I know that the cos(2π/3) because that's the x-value, and the x-value is negative, that's -1/2.0512

The sin(2π/3) is the y-value, which is positive, that's root 3 over 2.0524

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).0530

What we're doing there is working out each one of the formulas for sin(2x) and cos(2x) with x=π/3.0545

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/3.0556

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.0567

Our next example is to use the double angle formulas to prove a trigonometric identity.0577

It's not so obvious how to start with this one.0584

We're actually going to start with the right-hand side because it looks more complicated.0588

I'm going to start with the right-hand side and that's 2tan(x)/1+tan²(x).0592

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.0605

First thing, I'm going to do is to change everything into sines and cosines.0615

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.0619

If you got a tangent or a secant, or a cosecant or a cotangent, convert it into sines or cosines.0626

It will probably make your life easier.0631

I'll write this as 2, tangent, remember is sin/cos, and 1+tan², that's 1+sin²(x)/cos²(x).0633

Now, I see a lot of cosines in denominators here, I think we're going to try to clear those out.0651

We multiply top and bottom by cos²(x) and see what happens with that.0655

That's multiplying by 1, so that's safe.0662

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.0665

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).0677

Now, look at this, the top is exactly 2sin(x)×cos(x).0695

I remember that, that's my formula for sin(2x).0701

Now the bottom, that's the Pythagorean identity, so that's just 1, cos²+sin²(x) is 1.0709

This converts into sin(2x), but that's equal to the left-hand side of what we were trying to prove.0716

We started with the right-hand side because it looked a little more complicated there.0724

I see a bunch of tangents, I am not so sure what to do with those, I convert them into sines and cosines.0729

I see a lot of cosines in the denominator, so I multiply top and bottom by cos²(x).0736

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.0745

It reduces down into the right-hand side.0754

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).0760

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).0771

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.0780

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).0793

Now, I'm going to use, well, it's the addition and subtraction formulas but it's really the double angle formulas.0811

Of course, those come from the addition and subtraction formulas.0817

Now, sin(2x) is 2sin(x)×cos(x), that's the double angle formula for sine.0820

Of course, you find that out from the addition formula.0829

Cos(2x) is cos²(x)-sin²(x), that was the first double angle formula for cosine.0832

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).0844

Right now, I've got a bunch of cosines lying around, I'd like to move those down into the denominator.0852

The reason is because tangent is sin/cos, so I would like to be dividing by cosines.0858

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).0866

We're dividing top and bottom by cos²(x), that's dividing by 1, so that's legitimate, we'll see what happens.0876

Now, in the numerator, we get 2sin(x), we had a cos(x) before, we divided by cos², we get 2sin(x)/cos(x).0882

In the bottom, we're dividing everything by cos²(x), we get 1-sin²(x)/cos²(x).0896

That's really nice because now we have sin/cos everywhere and that's tangent.0908

We are asked to find everything in terms of tan(x).0913

What we get here is 2sin/cos is tan(x) over 1-sin²(x)/cos²(x) is tan²(x).0918

Our formula, our double angle formula for tangent is tan(2x)=2tan(x)/(1-tan²(x)).0931

Now, I didn't list this at the beginning of the lecture as one of the main formulas that you really need to memorize.0944

It kind of depends on your trigonometry class, in some classes they will ask you to memorize this formula, this formula for tan(2x).0949

I don't think it's worth memorizing.0957

In my trigonometry classes, I don't require my students to memorize these formulas for tan(2x).0960

I do require them to memorize sin(2x) and cos(2x) and I figure they can work out the other ones from that.0965

You may have a teacher who requires you to memorize the formula for tan(2x).0973

If so, here it is, here is the formula that you want to remember.0979

Let's check that out on a value that I already know the tangent of, let's try x=π/6.0984

The tan(2π/6), according to this formula, would be 2×tan(π/6)/(1-tan²(π/6)).0994

Now, π/6 is a common value, tan(π/6), I remember that, I've got that one memorized, it's root 3 over 3.1011

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/6.1023

You just divide them together and get the tan(π/6).1034

This is 2 times root 3 over 3, over 1 minus root 3 over 3 squared.1037

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 9.1048

That's 3/9 which is 1/3.1063

This is 2 root 3 over 3, divided by 2/3.1067

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).1076

Of course, 2π/6 is just π/3.1092

π/3 is another common value that I know the tangent of.1101

tan(π/3), I remember, is root 3, that's a common value.1105

Again, if you don't remember that, remember the sine and cosine of π/3, divide them together and you'll get root 3.1112

Look at that, our answers agree.1120

That confirms our formula for tan(2x).1122

To recap the important parts of that problem, we have to figure out tan(2x).1126

We wrote it as sin/cos of 2x.1132

We expanded each one of those using the double angle formulas that we learned at the beginning of the lesson.1135

Then, I was trying to get this in terms of tan(2x).1140

I wanted to get some cosines in the denominator, that's why I divided top and bottom by cos²(x).1144

That converted the thing into something in terms of tan(x).1150

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 3.1156

That checks the common value that I also know tan(π/3) is square root of 3.1170

We'll try some more examples of that later.1176