WEBVTT 00:00:06.861 --> 00:00:12.607 Let's take a look at double and half angle formulas. Here I've listed in this box the identities and formulas 00:00:12.609 --> 00:00:20.370 for sine 2 theta, cosine 2 theta, and tangent 2 theta. In order to get sine 2 theta, you could use the sum 00:00:20.372 --> 00:00:25.624 identity for sine alpha plus beta, where alpha and beta are both theta, and you would get this identity. 00:00:25.626 --> 00:00:32.134 The rest of these identities can be found in a similar way. These two in particular can be obtained by using 00:00:32.136 --> 00:00:38.138 the Pythagorean identity. So you need to be familiar with these identities. Let's look at an example where 00:00:38.140 --> 00:00:46.386 we use one. Here we want to evaluate the expression as the sine, cosine, or tangent of a double angle. 00:00:46.388 --> 00:00:53.903 That's our first step. Then we want to find the exact value without using a calculator. So look at this. 00:00:53.905 --> 00:01:02.427 You have cosine squared of 11 pi over 12 minus sine squared of 11 pi over 12. Which one does this look like? 00:01:02.429 --> 00:01:11.402 Well notice it looks like this one, where theta is 11 pi over 12. So it looks like cosine squared theta minus 00:01:11.404 --> 00:01:22.413 sine squared theta. So I can see that theta is 11 pi over 12. Now that I know that, this is equal to the 00:01:22.415 --> 00:01:33.653 cosine of 2 theta, and we just determined that theta was 11 pi over 12. Well if you reduce that, 00:01:33.655 --> 00:01:43.403 what do you get? That gives you 11 pi over 6. So we've taken this expression to values we couldn't find 00:01:43.405 --> 00:01:48.207 without a calculator to something that we know one of the exact values. This is one of the special 00:01:48.209 --> 00:01:55.041 angles that you should be familiar with. So here's the first part. We rewrote it as, in this case, the cosine of 00:01:55.043 --> 00:02:01.217 a double angle. And then do you remember what the cosine of 11 pi over 6 is? Where does the terminal 00:02:01.219 --> 00:02:13.969 side of 11 pi over 6 lie? Exactly, in quadrant four, and cosine in quadrant four is positive or negative? 00:02:13.971 --> 00:02:20.990 Positive, exactly. So this is the same as the cosine of pi over 6, and those should be values you know. 00:02:20.992 --> 00:02:28.247 It's the square root of 3 over 2. So first we wrote it as a double angle, cosine of a double angle, 00:02:28.249 --> 00:02:31.000 and then we found that value. 00:02:32.738 --> 00:02:37.757 Let's look at this example where we're given sine of theta is 5 over 7 and we know the terminal side of 00:02:37.759 --> 00:02:45.032 theta lies in quadrant two. We want to find the exact value of sine 2 theta, cosine 2 theta, and tangent 2 theta. 00:02:45.034 --> 00:02:50.750 Well the first thing we need to do is very important. We need to draw a picture for theta. We know the 00:02:50.752 --> 00:03:01.499 terminal side lies in quadrant two, so let's draw a picture. So here's my angle theta. We know that we 00:03:01.501 --> 00:03:10.583 need to come up with our coordinates. Well we know the sine of theta is 5 over 7, and remember that the 00:03:10.585 --> 00:03:24.052 sine of theta is defined to be y over r. So what I know is y is 5 and r is 7. What we need to figure out is x. 00:03:24.054 --> 00:03:29.558 Well we can easily use the Pythagorean theorem to find it, but we need to pay attention. So this point is 00:03:29.560 --> 00:03:43.062 x comma 5, and r is 7. In quadrant two, remember x is negative, so when we find x it's going to be negative 00:03:43.064 --> 00:03:58.305 the square root of 7 squared minus 5 squared. What does that give me? 49 minus 25 is 24, and remember 00:03:58.307 --> 00:04:05.096 we're always asked to simplify radicals if we can. This is the same thing as negative 2 square roots of 6. 00:04:05.098 --> 00:04:15.103 So now we know x, y, and r. This is very important. Always pay attention to whether x and y are 00:04:15.105 --> 00:04:21.610 positive or negative in your quadrants, and we know the sine of theta is 5 over 7, and we know the cosine 00:04:21.612 --> 00:04:28.871 of theta would be what? Well it's going to be x over r. It's going to be negative 2 square roots of 6 over 7. 00:04:28.873 --> 00:04:38.128 Tangent of theta would be y over x, or negative 5 over 2 square roots of 6, and let's go ahead and write 00:04:38.130 --> 00:04:45.136 our sine here. That was 5 over 7. So it's very important to do this part right because if you make a 00:04:45.138 --> 00:04:51.385 mistake here then the rest of it will be wrong. So now that we've got a picture, let's first find the sine of 00:04:51.387 --> 00:05:07.135 2 theta. Well the identity for that is 2 sine theta cosine theta, and now we just have to plug in our values. 00:05:07.137 --> 00:05:19.416 Sine of theta, 5 over 7, cosine of theta, negative 2 square root of 6 over 7, and let's see. What does that 00:05:19.418 --> 00:05:36.186 give me? Negative 20 square root of 6 over 49, and that can't be reduced. So that's my value for sine 2 theta. 00:05:36.188 --> 00:05:48.196 Now let's find cosine 2 theta. Well notice that there are three identities for cosine 2 theta. Since we were 00:05:48.198 --> 00:05:54.201 given the sine of theta, the easiest one to use is the identity that only involves sine of theta. So I will use 00:05:54.203 --> 00:06:03.968 1 minus 2 sine squared theta, and then we just have to plug in our value. Sine of theta remember was 00:06:03.970 --> 00:06:10.714 5 over 7. We want to square that, and remember your order of operations. So you want to square this. 00:06:10.716 --> 00:06:24.466 We get 25 over 49. That gives me 1 minus 50 over 49, and 1 is the same thing as 49 over 49, minus 50 over 49, 00:06:24.468 --> 00:06:41.255 gives me negative 1 over 49. So cosine 2 theta is negative 1 over 49. Now let's find the tangent of 2 theta. 00:06:41.257 --> 00:06:50.512 Well the identity is 2 tangent theta over 1 minus tangent squared theta. Well we need to plug in our 00:06:50.514 --> 00:07:03.763 values. Our tangent was, let's see. Where did we write it down? Negative 5 over 2 square roots of 6 over 00:07:03.763 --> 00:07:12.032 1 minus negative 5 over 2 square roots of 6 and then that's squared. So the first thing we need to do is 00:07:12.034 --> 00:07:22.034 we can reduce this to negative 5 over the square root of 6, and then we've got to square this. When you 00:07:22.036 --> 00:07:25.046 square this, remember you're going to square the numerator and the denominator. So you're going to get 00:07:25.048 --> 00:07:32.041 25 in the numerator. In the denominator, you're going to square 2 and get 4, square root of 6 [squared] is 6, 00:07:32.043 --> 00:07:46.796 4 times 6 is 24, and 1 minus 25 over 24, that's 24 over 24, so that's negative 1 over 24. We can't leave 00:07:46.798 --> 00:07:51.545 complex fractions. We're asked to simplify that. Remember that's the same thing as negative 5 over the 00:07:51.547 --> 00:08:02.580 square root of 6 times the reciprocal, times negative 24, which gives me a positive number. I have a 00:08:02.582 --> 00:08:13.844 negative times a negative. That's 120 over the square root of 6. So that's my value for tangent 2 theta. 00:08:13.846 --> 00:08:21.845 One comment about tangent 2 theta. Since we found the sine of 2 theta and the cosine of 2 theta, if you 00:08:21.847 --> 00:08:32.615 knew both of those you could have found the tangent 2 theta by doing sine 2 theta over cosine 2 theta. 00:08:32.617 --> 00:08:41.116 You already had those values. So if you took negative 20 square roots of 6 over 49 divided by 00:08:41.118 --> 00:08:52.870 cosine, which was negative 1 over 49, this would simplify to positive 20 square roots of 6. The 49's will 00:08:52.872 --> 00:08:59.112 divide out. These two answers are equivalent. So if you don't know the sine 2 theta and cosine 2 theta, 00:08:59.114 --> 00:09:03.862 then you would just go ahead and use this formula. If you know them then of course you have a choice. 00:09:06.140 --> 00:09:11.144 Let's take a look at the half-angle formulas. Here I've listed on the board all the half-angles for sine alpha 00:09:11.146 --> 00:09:20.146 over 2, cosine alpha over 2, tangent alpha over 2. Notice for sine alpha over 2 there are two formulas, one 00:09:20.148 --> 00:09:27.896 if alpha over 2, the terminal side, lies in quadrant one or two and negative square root of 1 minus cosine 00:09:27.898 --> 00:09:33.176 alpha over 2 when alpha over 2, the terminal side, lies in quadrant three or four. So it's very important 00:09:33.178 --> 00:09:38.929 to first know where alpha over 2 is and pick the correct formula. The same thing for cosine and tangent. 00:09:38.931 --> 00:09:44.683 Although for tangent, notice the last two. Here, same thing, you'd have to know where alpha over 2 lies, 00:09:44.685 --> 00:09:50.692 but the last two, alpha over 2 can be in any quadrant. Let's look at an example. We want to use a 00:09:50.694 --> 00:09:57.192 half-angle formula to evaluate cosine 7 pi over 8 without using a calculator. Notice 7 pi over 8 is not a 00:09:57.194 --> 00:10:08.192 special angle, so it says use a half-angle formula. That means that 7 pi over 8 is the half-angle, so alpha 00:10:08.194 --> 00:10:16.698 over 2, and from this we can determine what alpha is. If we multiply both sides by 2, then we see that alpha 00:10:16.700 --> 00:10:28.232 is 7 pi over 4. So we want to find the exact value of [cosine] 7 pi over 8. So we've determined alpha. We have 00:10:28.234 --> 00:10:36.489 to choose the formula. So where does 7 pi over 8 lie? Think about that. 7 pi over 8 is a little less than pi, 00:10:36.491 --> 00:10:43.743 so it's in quadrant two. Cosine in quadrant two is negative, so we would choose this formula. So you write 00:10:43.745 --> 00:10:53.752 down the formula. It's [negative square root of] 1 plus cosine alpha over 2. A lot of students make the 00:10:53.754 --> 00:10:59.768 mistake of choosing the sign based on alpha. Don't do that. You always choose the formula based on where 00:10:59.770 --> 00:11:06.510 alpha over 2 is. Always write down the formula, and then plug in your value. We found alpha to be 00:11:06.512 --> 00:11:16.256 7 pi over 4, so we want the cosine of 7 pi over 4. Don't skip any steps, and then you have to find the cosine 00:11:16.258 --> 00:11:28.619 of 7 pi over 4. Well 7 pi over 4 lies in what quadrant? 7 pi over 4 is in quadrant four. Cosine in quadrant four 00:11:28.621 --> 00:11:39.795 is positive. So this means this is going to be positive. Cosine of 7 pi over 4 is the same thing as cosine of 00:11:39.797 --> 00:11:50.807 pi over 4, which is 1 over the square root of 2. Now we can't leave complex fractions. We need to simplify them. 00:11:50.809 --> 00:11:56.078 In order to do that, what you're going to do is you're going to multiply underneath the radical by the 00:11:56.080 --> 00:12:03.075 square root of 2 over the square root of 2, and so what does that give me? Well, 1 times the square root 00:12:03.077 --> 00:12:09.323 of 2 you get the square root of 2, plus 1, in the numerator, and in the denominator you get 2 square roots 00:12:09.325 --> 00:12:21.573 of 2. So the cosine of 7 pi over 8 is negative square root of square root of 2 plus 1 over 2 square roots of 2. 00:12:23.070 --> 00:12:28.576 In this example, we're given the tangent of alpha is 5 over 12. We're told that alpha is between pi and 00:12:28.578 --> 00:12:34.586 3 pi over 2, and we want to determine the exact values of sine of alpha over 2, cosine alpha over 2, 00:12:34.588 --> 00:12:41.575 tangent alpha over 2. Well the first thing we need to do is we need to draw a picture for alpha. We know 00:12:41.577 --> 00:12:47.571 the terminal side lies in quadrant? If we're between pi and 3 pi over 2 that means it lies in quadrant three. 00:12:47.573 --> 00:13:06.078 So let's draw a picture for alpha. We know the tangent of alpha is 5 over 12, and remember tangent is 00:13:06.080 --> 00:13:14.072 defined to be y over x. You might be tempted to think both x and y are positive, but they're not. Remember, 00:13:14.074 --> 00:13:23.159 you're in quadrant three, and in quadrant three x and y are both negative. So that means that x is negative 12 00:13:23.159 --> 00:13:34.417 and y is negative 5. We use the Pythagorean theorem. We get r to be 13. So now that we know our x, y, and r, 00:13:34.419 --> 00:13:45.915 we can find, for example, what's the cosine of alpha? Well it would be negative 12 over 13. Sine of alpha is 00:13:45.917 --> 00:13:55.671 negative 5 over 13, and tangent we already know. Tangent is 5 over 12. Ok, so now that we know this, let's 00:13:55.673 --> 00:14:08.650 start with the sine of alpha over 2. Well the first thing we have to do is we have to pick a formula, and 00:14:08.652 --> 00:14:16.450 remember, we pick the formula based on where alpha over 2 is. This is a picture for alpha. Where's alpha over 2? 00:14:16.452 --> 00:14:24.706 Well what we can do is take this expression. We know that alpha is between pi and 3 pi over 2. If we 00:14:24.708 --> 00:14:33.709 divide this inequality by 2, that will tell us where the terminal side of alpha over 2 lies, so we can see 00:14:33.711 --> 00:14:41.227 that the terminal side of alpha over 2 is between pi over 2 and 3 pi over 4. What quadrant is that? 00:14:41.229 --> 00:14:49.494 Quadrant two. So that's how we decide. We know that the terminal side of alpha over 2 lies in quadrant two. 00:14:49.496 --> 00:15:01.993 We want a sine, and sine in quadrant two is positive, so we choose this formula. Write down your formula. 00:15:01.995 --> 00:15:15.987 Don't skip any steps, and then plug in your value. Cosine of alpha is negative 12 over 13, so underneath the 00:15:15.989 --> 00:15:28.487 radical you have 1 plus 12/13. Well 1 is 13/13 plus 12/13 gives you 25/13, and that's the same thing as 00:15:28.489 --> 00:15:49.300 25 over 26, which can be simplified because the square root of 25 is 5. So the sine of alpha over 2 I get to be 00:15:49.302 --> 00:16:00.048 5 over the square root of 26. Now let's find the cosine of alpha over 2. Well we've already determined, remember 00:16:00.050 --> 00:16:05.566 that alpha over 2, the terminal side, lies in quadrant two, and cosine in quadrant two, is that positive or negative? 00:16:05.568 --> 00:16:15.573 Negative, exactly, so you're going to use the negative square root of 1 plus cosine alpha over 2, 00:16:15.575 --> 00:16:29.835 and then you plug in your value for cosine. Let's see I'll go this way. Cosine of alpha was negative 12 over 13, 00:16:29.837 --> 00:16:40.838 and so what is that? That's 13 over 13 minus 12, so that's 1 over 13 over 2, which is the same thing as 00:16:40.840 --> 00:16:55.112 1 over 26, and the square root of 1 of course is 1. So this is negative 1 over the square root of 26. 00:16:55.114 --> 00:17:05.870 So now we've found the cosine of alpha over 2. Last, let's find the tangent of alpha over 2. Well since 00:17:05.872 --> 00:17:10.884 we have a choice on this identity, let's use one where we don't have to choose whether it's positive 00:17:10.886 --> 00:17:21.389 or negative. Let's use this one, 1 minus cosine alpha over sine alpha, and then plug in our values. 00:17:21.391 --> 00:17:32.903 Cosine of alpha was negative 12 over 13. Sine of alpha was negative 5 over 13. So again this is the same 00:17:32.905 --> 00:17:45.170 thing as 1 plus 12 /13 which is 13 over 13 plus 12, so that give me 25 over 13 over negative 5 over 13. 00:17:45.172 --> 00:17:56.176 That's the same thing as 25 over 13 times negative 13 over 5, and that gives you what? Negative 5 00:17:56.178 --> 00:18:09.426 for tangent alpha over 2. So the tangent of alpha over 2 is negative 5. Let me point out, it's very important, 00:18:09.428 --> 00:18:14.926 this part of the step is very important. If you draw the picture wrong, if you don't pay attention to x, y, 00:18:14.928 --> 00:18:18.701 whether they're positive or negative, you'll make a mistake, and of course all this will be wrong. 00:18:18.703 --> 00:18:25.206 Pay special attention when you're drawing your picture. Make sure you choose the right formula, 00:18:25.208 --> 00:18:29.700 based on where alpha over 2 lies. Make sure you plug in the values. 00:18:30.958 --> 00:18:37.214 In this example, we want to find the exact value without using a calculator. So look at this first example. 00:18:37.216 --> 00:18:46.223 We have the sine of 2 inverse cosine of negative 1/3. First we want to look at this, the inverse cosine of 00:18:46.225 --> 00:18:50.476 negative 1/3, and we're going to think of that as an angle. We're going to call it theta, so we're going to 00:18:50.478 --> 00:19:00.226 let theta be the inverse cosine of negative 1/3. Which means if you rewrite that, that's the same thing as the cosine of 00:19:00.228 --> 00:19:08.244 theta is negative 1/3. But remember your range for inverse cosine. Where does this angle lie? What interval? 00:19:08.246 --> 00:19:19.246 Do you remember? Yes, 0 to pi, so theta must be between 0 and pi, and we know the cosine is negative. 00:19:19.248 --> 00:19:25.496 So where is it? That's either quadrant one or quadrant two, so we know theta lies, the terminal side, in 00:19:25.498 --> 00:19:35.992 quadrant two. So we can draw a picture for this angle. Well since we know the cosine is negative 1 over 3, 00:19:35.994 --> 00:19:45.995 we know cosine is x over r. So I know x is negative 1. I know r is 3. We need to find y. Now notice in the 00:19:45.997 --> 00:19:54.292 second quadrant the y-coordinate is positive, so it's going to be positive square root of 9 minus 1, which is 00:19:54.294 --> 00:20:04.794 square root of 8, which simplifies to 2 square roots of 2. So now we've found x, y, and r. Knowing that, 00:20:04.796 --> 00:20:15.813 let's go back to this problem. We have the sine of 2 theta. Well we can use an identity. Remember the 00:20:15.815 --> 00:20:25.596 identity for sine 2 theta? It's 2 sine theta cosine theta. So we need to know the sine of theta and the cosine 00:20:25.598 --> 00:20:31.325 of theta. Well, we already know the cosine of theta is negative 1/3. Since we found y, what would the 00:20:31.327 --> 00:20:43.090 sine of theta be? It would be 2 square roots of 2 over 3. So we can plug in those values, 2 square roots of 2 00:20:43.092 --> 00:20:58.590 over 3 for sine theta, cosine negative 1/3, and so I get a negative 4 square roots of 2 over 9 for sine 2 theta. 00:20:58.592 --> 00:21:12.340 Let's scoot this over. So once again, it's important to draw the picture. You have to remember your range 00:21:12.342 --> 00:21:19.124 for inverse cosine and draw it in the correct quadrant. So now we know sine 2 theta is negative 4 square roots 00:21:19.126 --> 00:21:26.632 of 2 over 9. Let's look at another example. We want to find the exact value here as well. We see the sine of 00:21:26.634 --> 00:21:33.154 1/2 inverse tangent of negative 1. Well once again, look at this, inverse tangent of negative 1. Think of it as 00:21:33.156 --> 00:21:40.895 as an angle. We're going to call it alpha. We're going to let alpha be the inverse tangent of negative 1, and 00:21:40.897 --> 00:21:46.153 again, what does that mean? It means the tangent of alpha is negative 1, and again, you have to remember 00:21:46.155 --> 00:21:52.403 the range for inverse tangent. It's very important. Do you remember what the range is? Alpha has to be 00:21:52.405 --> 00:21:59.918 between where? Negative pi over 2 and pi over 2, which means the terminal side either lies in quadrant 00:21:59.920 --> 00:22:08.168 four or quadrant one. Well, we know the tangent is negative, so we know that the terminal side lies in 00:22:08.170 --> 00:22:20.186 quadrant four, and remember the tangent is y over x, so we're going to have to figure out our coordinates 00:22:20.188 --> 00:22:32.701 again. Well, let's see, in quadrant four, which value is negative? The y, right. So we know y is negative. 00:22:32.703 --> 00:22:40.454 Let y be negative 1 and x be 1, so the tangent is negative 1. Use the Pythagorean theorem. R is going to be 00:22:40.456 --> 00:22:52.969 the square root of 2. So now we know cosine of alpha is 1 over the square root of 2, and if we need the sine, 00:22:52.971 --> 00:22:59.976 it would be negative 1 over the square root of 2. So we've got all our values. We've paid attention to what 00:22:59.976 --> 00:23:07.225 quadrant, and now we're ready to go back to here. So we want to know the sine of 1/2 alpha. That's the 00:23:07.227 --> 00:23:19.747 same thing as the sine of alpha over 2. Remember there are two formulas for the sine of alpha over 2. 00:23:19.749 --> 00:23:35.496 There's this one, and then there's this one. Oops, negative. And remember, you choose this based on 00:23:35.498 --> 00:23:42.519 where alpha over 2 is. Well we've drawn a picture for alpha, not alpha over 2. But notice, where's alpha? 00:23:42.521 --> 00:23:52.020 We said it's in quadrant four. So we actually know that alpha lies between negative pi over 2 and 0 because 00:23:52.022 --> 00:23:59.036 we've narrowed it down to just quadrant four. We can divide this inequality by 2 to figure out where the 00:23:59.038 --> 00:24:06.534 terminal side of alpha over 2 lies. So let's see. Alpha over 2 is between negative pi over 4 and 0, so that 00:24:06.536 --> 00:24:13.286 means that the terminal side of alpha over 2 lies also in quadrant four. Remember, sine in quadrant 00:24:13.288 --> 00:24:21.285 four is, which one do I use, positive or negative? Quadrant four sine is negative, so I would use this one. 00:24:21.287 --> 00:24:28.785 Write it down here. So it's going to be [negative square root of] 1 minus cosine alpha over 2, and again 00:24:28.787 --> 00:24:37.322 plug in your value for cosine alpha, which is 1 over the square root of 2. Once again, since we have 00:24:37.324 --> 00:24:43.079 a complex fraction, we need to simplify by multiplying by the square root of 2 over the square root of 2 00:24:43.081 --> 00:24:51.580 underneath the radical. So the sine of this alpha over 2 is what? Let's see. It would be the square root of 2 00:24:51.582 --> 00:25:00.330 minus 1 over 2 square roots of 2. So again, it's very important that you know the range for the inverse 00:25:00.332 --> 00:25:05.852 functions, that you draw the picture, you pay attention to whether x and y are positive or negative, and 00:25:05.854 --> 00:25:10.601 don't skip any steps. I see students skipping steps, and they make a mistake. So make sure you write down 00:25:10.603 --> 00:25:16.851 the formula and then plug in your values. Practice working these examples until you've mastered the 00:25:16.853 --> 00:25:19.850 skills and concepts involved.