1 00:00:01,450 --> 00:00:05,002 In this video, we're going to be looking at the double angle 2 00:00:05,002 --> 00:00:11,130 formula. But to start with, we're going to start from the 3 00:00:11,130 --> 00:00:16,696 addition formula. Not all of them, just the ones that deal 4 00:00:16,696 --> 00:00:22,888 with A+B. So let's just write those down to begin with sign of 5 00:00:22,888 --> 00:00:25,396 A+B, we know is sign a. 6 00:00:25,950 --> 00:00:32,838 Cause B. Post cause a sign 7 00:00:32,838 --> 00:00:40,310 be. Next, we want the cause of A+B, which 8 00:00:40,310 --> 00:00:44,260 will be cause a calls 9 00:00:44,260 --> 00:00:51,387 B. Minus sign, a sign be and finally 10 00:00:51,387 --> 00:00:57,795 the tan one tan of A+B, which will 11 00:00:57,795 --> 00:01:00,999 be 10 A plus 12 00:01:00,999 --> 00:01:06,629 10B. Over 1 - 13 00:01:06,629 --> 00:01:10,688 10 a 10B. 14 00:01:12,290 --> 00:01:15,394 So those are three of our addition formula. 15 00:01:16,680 --> 00:01:20,088 And each one is to do with A+B. 16 00:01:20,990 --> 00:01:27,843 So what happens if we let a be equal to be? 17 00:01:28,480 --> 00:01:34,161 In other words, instead of having a plus B, we have a plus 18 00:01:34,161 --> 00:01:41,760 a. So that would be sign of A plus a would 19 00:01:41,760 --> 00:01:44,252 be signed to A. 20 00:01:45,720 --> 00:01:52,970 What does that do to this right hand side? Well 21 00:01:52,970 --> 00:02:00,220 gives us sign a cause, A plus cause a sign 22 00:02:00,220 --> 00:02:08,157 a. In other words, these two at the same, so we can just add 23 00:02:08,157 --> 00:02:15,642 them together. So sign of 2A is 2 sign a cause A and that's our 24 00:02:15,642 --> 00:02:20,133 first double angle formula double angle because it's 2A 25 00:02:20,133 --> 00:02:22,129 where doubling the angle. 26 00:02:22,670 --> 00:02:29,703 So what is it sign and so on. Let's do the same with 27 00:02:29,703 --> 00:02:36,736 cause. Let's put a equal to be. So will have cars to a 28 00:02:36,736 --> 00:02:42,687 is equal to what it was cause a Cosby. It's now 29 00:02:42,687 --> 00:02:48,638 going to be cause A cause a witches caused squared A. 30 00:02:49,720 --> 00:02:56,902 Sign a sign be when it's now going to be sign a sign a 31 00:02:56,902 --> 00:02:59,980 which is sine squared minus sign 32 00:02:59,980 --> 00:03:03,428 squared A. And that's how 33 00:03:03,428 --> 00:03:07,250 a second. Double angle formula. 34 00:03:07,930 --> 00:03:15,290 Doing the same with Tan Tan 2A is equal to. 35 00:03:15,940 --> 00:03:23,738 10A plus 10 B this is now angle a so it's 10A Plus 10A 36 00:03:23,738 --> 00:03:26,523 which is 2 Tab A. 37 00:03:26,540 --> 00:03:33,356 All over 1 - 10, eight and be. But this is now a instead of B, 38 00:03:33,356 --> 00:03:34,634 so it's tanae. 39 00:03:35,290 --> 00:03:41,167 10 eight times by 10 A is 10 squared. 40 00:03:41,167 --> 00:03:44,432 1 - 10 squared A. 41 00:03:46,150 --> 00:03:51,217 And here are our three double angle formula again 42 00:03:51,217 --> 00:03:56,847 to be learned to be recognized and to be used. 43 00:03:58,680 --> 00:04:04,477 Let's just have a look at this one cause to A. 44 00:04:05,750 --> 00:04:10,310 White pick out this one. Well this right hand side which is 45 00:04:10,310 --> 00:04:14,490 the bit that interests because it's got cost squared and sign 46 00:04:14,490 --> 00:04:17,150 squared in it and there is an 47 00:04:17,150 --> 00:04:21,130 identity. That's to do with cost squared. Plus sign squared 48 00:04:21,130 --> 00:04:26,800 equals 1. What that means is we can replace the sine squared. 49 00:04:27,340 --> 00:04:32,016 And get everything in terms of Cos squared. Or we can do it the 50 00:04:32,016 --> 00:04:33,018 other way round. 51 00:04:33,600 --> 00:04:36,903 So I just have a look at that one. 52 00:04:38,280 --> 00:04:45,222 Cause to a cost squared, A 53 00:04:45,222 --> 00:04:48,693 minus sign squared 54 00:04:48,693 --> 00:04:55,745 a butt. Cost squared A plus sign, 55 00:04:55,745 --> 00:04:59,517 squared A equals 1. 56 00:05:00,200 --> 00:05:07,337 In other words, sign squared a is 1 minus Cos squared a so 57 00:05:07,337 --> 00:05:13,925 we can replace the sine squared here in our double angle formula 58 00:05:13,925 --> 00:05:19,964 by one minus Cos squared, so will have cause to A. 59 00:05:20,560 --> 00:05:28,140 Is cost squared A minus one minus Cos squared A? 60 00:05:29,150 --> 00:05:34,160 Using the brackets, notice to show I'm taking away all 61 00:05:34,160 --> 00:05:38,168 of it and now let's remove the brackets. 62 00:05:39,370 --> 00:05:46,630 Minus one. Minus minus gives me a plus 63 00:05:46,630 --> 00:05:53,330 cause squared a, so I now have two cost squared, 64 00:05:53,330 --> 00:06:00,030 A minus one, so that's another double angle formula for 65 00:06:00,030 --> 00:06:02,040 cost to a. 66 00:06:02,770 --> 00:06:09,346 Now because I replaced the sine squared here by one minus Cos 67 00:06:09,346 --> 00:06:16,470 squared, I can do the same again and replace the cost squared by 68 00:06:16,470 --> 00:06:23,594 one minus sign squared and what that will give main is cause 2A 69 00:06:23,594 --> 00:06:27,430 is 1 - 2 sine squared A. 70 00:06:27,460 --> 00:06:35,100 So lot of formally there. Let's just write them all 71 00:06:35,100 --> 00:06:42,306 down again. Sign to a 72 00:06:42,306 --> 00:06:46,210 IS2. Find a 73 00:06:46,210 --> 00:06:50,163 Kohl's A. Calls 74 00:06:50,163 --> 00:06:56,884 to a. Is cost squared A minus sign, 75 00:06:56,884 --> 00:07:03,674 squared A and we can rewrite that as two cost 76 00:07:03,674 --> 00:07:10,464 square day minus one or as 1 - 2 sine 77 00:07:10,464 --> 00:07:18,212 squared A. And then turn to a is 78 00:07:18,212 --> 00:07:21,128 equal to 2. 79 00:07:21,130 --> 00:07:27,538 Tam a over 1 - 10 80 00:07:27,538 --> 00:07:33,264 squared A. So there are our double angle formula 81 00:07:33,264 --> 00:07:37,360 formula to be learned formally to be remembered 82 00:07:37,360 --> 00:07:40,944 and most importantly recognized and used when 83 00:07:40,944 --> 00:07:42,480 we need them. 84 00:07:43,840 --> 00:07:49,566 So let's have a look at how we can make use of these double 85 00:07:49,566 --> 00:07:57,260 angle formula. So sign of three X. Is it possible to write 86 00:07:57,260 --> 00:08:02,957 sign of 3X all in terms of sine X? 87 00:08:04,480 --> 00:08:07,805 Well. Let's try and break this 88 00:08:07,805 --> 00:08:15,319 3X up. 3X is 2X Plus X, so we can write this a sign of 89 00:08:15,319 --> 00:08:16,738 2X Plus X. 90 00:08:18,230 --> 00:08:24,970 OK, this means we can use our addition formula sign 91 00:08:24,970 --> 00:08:28,340 of two X cause X. 92 00:08:28,930 --> 00:08:35,566 Plus cause of two X sign X. 93 00:08:36,550 --> 00:08:44,002 Now I can use my double angle formula here sign of two 94 00:08:44,002 --> 00:08:51,454 X is 2 sign X Cos X still to be multiplied by 95 00:08:51,454 --> 00:08:53,317 Cos X Plus. 96 00:08:54,040 --> 00:08:56,730 Now I have a choice. 97 00:08:57,380 --> 00:09:02,418 There are three double angle formula for cause 2X, so my 98 00:09:02,418 --> 00:09:09,288 choice is got to be governed by what it is I'm trying to do and 99 00:09:09,288 --> 00:09:14,784 we're trying to write sign 3X all in terms of sine X. 100 00:09:15,400 --> 00:09:22,218 That means the choice I have to make here is the one that's got 101 00:09:22,218 --> 00:09:28,062 signs in it, not cosines, but the one that's got signs and 102 00:09:28,062 --> 00:09:35,367 only signs, and the one that has that is 1 - 2 sine squared X 103 00:09:35,367 --> 00:09:38,776 still to be times by sign X. 104 00:09:39,580 --> 00:09:46,830 So this front term is going to be 2 sign 105 00:09:46,830 --> 00:09:49,730 X cause squared X. 106 00:09:50,300 --> 00:09:57,830 One times by Cynex is plus sign X minus and 107 00:09:57,830 --> 00:10:05,360 two sine squared X times. Biosynex is sine cubed X. 108 00:10:06,320 --> 00:10:11,204 Well, we're getting there. We've got sign here sign here. Sign 109 00:10:11,204 --> 00:10:13,424 cubed here. Cost squared here. 110 00:10:14,340 --> 00:10:19,940 But Cost Square can be rewritten using one of the fundamental 111 00:10:19,940 --> 00:10:24,637 identity's cost square plus sign squared is one so cost square 112 00:10:24,637 --> 00:10:26,772 can be replaced by Wang. 113 00:10:27,630 --> 00:10:32,850 Minus sign squared. 114 00:10:34,940 --> 00:10:40,256 And so we can see here. Everything is now in terms of 115 00:10:40,256 --> 00:10:44,243 sine X and all we need to do is 116 00:10:44,243 --> 00:10:51,580 tidied up. So we multiply out this bracket 2 sign X for 117 00:10:51,580 --> 00:10:58,540 the first term, 2 sign X times by one. Then we have 118 00:10:58,540 --> 00:11:04,920 two sign X times Y minus sign squared minus two sine 119 00:11:04,920 --> 00:11:10,720 cubed X plus sign X minus two sine cubed X. 120 00:11:11,730 --> 00:11:18,920 2 sign X plus sign X that's three sign X. 121 00:11:19,770 --> 00:11:27,366 Minus two sine cubed minus two sine cubed is minus 4 sign 122 00:11:27,366 --> 00:11:34,329 cubed X and that everything is in terms of sine X. 123 00:11:35,340 --> 00:11:41,598 You can do the same with cause as well cause 3X can be turned 124 00:11:41,598 --> 00:11:46,068 into an expression that's entirely in terms of cause X. 125 00:11:46,830 --> 00:11:52,386 That's an example of using our double angle formula in order to 126 00:11:52,386 --> 00:11:57,479 reduce if we like to use that expression and multiple angle 127 00:11:57,479 --> 00:12:03,961 sign 3X is a multiple angle down to a single angle in terms of 128 00:12:03,961 --> 00:12:09,980 the sign of that angle. Let's have a look now at solving an 129 00:12:09,980 --> 00:12:17,063 equation. Let's take cause 2X is equal to 130 00:12:17,063 --> 00:12:23,535 sign X and let's take a range of 131 00:12:23,535 --> 00:12:25,962 values for X. 132 00:12:26,540 --> 00:12:31,980 Which puts X between plus and minus pie. 133 00:12:33,590 --> 00:12:36,622 Again, I've deliberately chosen 134 00:12:36,622 --> 00:12:42,806 caused 2X. Be cause we have a choice, we have three 135 00:12:42,806 --> 00:12:45,076 possibilities. Which one do we 136 00:12:45,076 --> 00:12:50,640 choose? Well, if I want to solve an equation like this, I really 137 00:12:50,640 --> 00:12:53,680 need it all in terms of one trig 138 00:12:53,680 --> 00:12:56,740 function. Not two, but one. 139 00:12:58,050 --> 00:13:03,650 And here I've got sine X. Therefore makes sense here. 140 00:13:04,240 --> 00:13:10,696 To replace this by 1 - 2 sine 141 00:13:10,696 --> 00:13:13,924 squared, X equals sign 142 00:13:13,924 --> 00:13:19,866 X. Now we have a quadratic equation where the 143 00:13:19,866 --> 00:13:22,134 variable is sign X. 144 00:13:23,100 --> 00:13:26,898 Let's rearrange that so that it 145 00:13:26,898 --> 00:13:32,260 equals 0. Add the two sine squared to each side. 146 00:13:33,680 --> 00:13:36,875 Plus the sign 147 00:13:36,875 --> 00:13:41,799 X. And take one away from each side. 148 00:13:43,050 --> 00:13:47,200 This is now a quadratic equation. Can I factorize it? 149 00:13:47,200 --> 00:13:48,860 Let's have a look. 150 00:13:48,860 --> 00:13:54,283 Two brackets, 2 sign X and sign X when multiplied together, 151 00:13:54,283 --> 00:14:00,692 these two will give me the two sine squared I need minus one, 152 00:14:00,692 --> 00:14:07,101 so let's pop a one into each bracket, and one of them's got 153 00:14:07,101 --> 00:14:10,059 to be plus and one minus. 154 00:14:11,240 --> 00:14:17,645 I need plus sign X in the middle going to make this one plus one, 155 00:14:17,645 --> 00:14:24,477 so I get +2 sign X, make that one minus so I get minus sign X 156 00:14:24,477 --> 00:14:29,174 and when I combine those two terms plus sign X there. 157 00:14:30,470 --> 00:14:37,098 This says. A bracket, a lump of algebra times by 158 00:14:37,098 --> 00:14:41,608 another bracket. Another lump of algebra is equal to 0. 159 00:14:42,630 --> 00:14:44,238 And so one. 160 00:14:44,850 --> 00:14:51,738 Or both of these brackets must be equal 161 00:14:51,738 --> 00:14:58,108 to 0. And so we've reduced this fairly complicated 162 00:14:58,108 --> 00:15:05,412 looking equation down to two simple ones, and this one tells 163 00:15:05,412 --> 00:15:13,380 us here. Sign X is equal to add 1 to each side 164 00:15:13,380 --> 00:15:16,036 and divide by two. 165 00:15:16,130 --> 00:15:21,915 Sign X is 1/2 or this one here tells us that sign X 166 00:15:21,915 --> 00:15:24,140 is equal to minus one. 167 00:15:25,700 --> 00:15:29,935 We've now got to extract the values of X from this 168 00:15:29,935 --> 00:15:34,555 information and those values of X must be between plus and minus 169 00:15:34,555 --> 00:15:41,300 pie. So let's sketch the graph of cynex between 170 00:15:41,300 --> 00:15:43,980 plus and minus pie. 171 00:15:45,310 --> 00:15:48,150 There's the graph, there's pie. 172 00:15:48,950 --> 00:15:55,880 Pie by 2 - π by two and minus pie and it goes between 173 00:15:55,880 --> 00:16:02,810 one and minus one. So let's take this one. First sign X is minus 174 00:16:02,810 --> 00:16:09,740 one. Well that goes across there and down to their, so X is minus 175 00:16:09,740 --> 00:16:13,700 π by two is one answer that we 176 00:16:13,700 --> 00:16:20,171 get there. Sign X is 1/2, half goes across there and we should 177 00:16:20,171 --> 00:16:25,343 recognize that this is one of those nice numbers. Sign X is 178 00:16:25,343 --> 00:16:31,377 1/2 for which we've got an exact answer, and So what we do know 179 00:16:31,377 --> 00:16:36,980 is that the sign of 30 degrees is 1/2, but where working in 180 00:16:36,980 --> 00:16:43,014 radians. So in fact 30 degrees is the same angle as pie by 6. 181 00:16:43,520 --> 00:16:47,870 And this is symmetrical. Remember the curve for sign is 182 00:16:47,870 --> 00:16:53,960 symmetrical, so if that's pie by 6 in there, that's got to be pie 183 00:16:53,960 --> 00:17:01,355 by 6 in there. So this, In other words will be 5 pie by 6, and so 184 00:17:01,355 --> 00:17:08,315 we have our two answers for this one pie by 6 and five pie by 6. 185 00:17:08,350 --> 00:17:14,447 So that we see that we've been able to solve our equation using 186 00:17:14,447 --> 00:17:19,137 our double angle formula and by making the right choice, 187 00:17:19,137 --> 00:17:24,765 particularly here when with cost 2X, we know that we have three 188 00:17:24,765 --> 00:17:31,562 possibilities. So. That is, have a look at another 189 00:17:31,562 --> 00:17:37,568 equation again using our double angle formula. Sign 2 X equals 190 00:17:37,568 --> 00:17:44,666 sign X. And again, let's take our value of X to lie between 191 00:17:44,666 --> 00:17:46,850 plus and minus pie. 192 00:17:48,160 --> 00:17:53,128 We've only one choice for sign 2X, that's two. 193 00:17:53,730 --> 00:18:00,604 Sign X Cos X equals sign X. 194 00:18:01,500 --> 00:18:07,604 Now. It's very, very tempting to say our common factor on each 195 00:18:07,604 --> 00:18:09,180 side. Cancel it out. 196 00:18:10,500 --> 00:18:11,860 And then we've lost it. 197 00:18:12,710 --> 00:18:18,446 And because we lose it, we might lose solutions to the equation. 198 00:18:19,120 --> 00:18:25,511 So what's better than canceling out is to get everything to 199 00:18:25,511 --> 00:18:32,756 one side. By taking cynex away from each side. 200 00:18:33,310 --> 00:18:39,770 And then. Take out a common factor, and here there's 201 00:18:39,770 --> 00:18:43,160 a common factor of sign X. 202 00:18:43,730 --> 00:18:50,530 Which will leave us 2 cause AX minus 203 00:18:50,530 --> 00:18:58,037 one. Two expressions multiplied together give us 0. 204 00:18:58,960 --> 00:19:06,520 So either or both of these expressions is equal to 0, so 205 00:19:06,520 --> 00:19:12,190 either sign X equals 0 or two cause X. 206 00:19:12,830 --> 00:19:15,518 Minus one equals 0. 207 00:19:16,330 --> 00:19:22,358 Now we've managed to reduce this equation to two smaller, simpler 208 00:19:22,358 --> 00:19:27,290 equations once sign and the other ones for cause. 209 00:19:27,820 --> 00:19:33,115 But each is going to give us a value of X. Let's take this one 210 00:19:33,115 --> 00:19:35,233 first, sign of X is 0. 211 00:19:36,290 --> 00:19:43,670 And let's draw a sketch up here of sign of X. There 212 00:19:43,670 --> 00:19:51,050 it is and it's zero here, here and here on the X 213 00:19:51,050 --> 00:19:57,815 axis minus Π Zero and Pi straightaway. We've got X equals 214 00:19:57,815 --> 00:20:05,195 minus π and 0, not pie, because pie is excluded from the 215 00:20:05,195 --> 00:20:07,070 range that. We've got. 216 00:20:07,580 --> 00:20:14,255 But notice sign X equals 0 gave us 2 answers for X if we have 217 00:20:14,255 --> 00:20:20,930 cancelled sign X out up here and just got rid of it, we would not 218 00:20:20,930 --> 00:20:23,155 have got those two answers. 219 00:20:23,770 --> 00:20:28,582 Let's go to this equation now. 2 cause X minus one is 220 00:20:28,582 --> 00:20:33,394 0, so that tells us that cause X is equal to 1/2. 221 00:20:34,540 --> 00:20:39,448 And what we need to do is sketch the graph of cosine. 222 00:20:40,430 --> 00:20:42,490 And the graph of cosine. 223 00:20:43,500 --> 00:20:50,250 Looks. Like that between pie and minus pie? 224 00:20:51,130 --> 00:20:54,620 And here is the half. 225 00:20:56,170 --> 00:21:02,338 And cause X equals 1/2. This is another one of these nice 226 00:21:02,338 --> 00:21:07,992 relationships and we know that this one is 60 degrees or 227 00:21:07,992 --> 00:21:13,646 because we're working in radians Pi by three and because of 228 00:21:13,646 --> 00:21:20,328 symmetry this one here has got to be minus π by three, so 229 00:21:20,328 --> 00:21:25,468 X is minus π by 3 or π by 3. 230 00:21:25,590 --> 00:21:31,530 And so we've got our four solutions for this equation. 231 00:21:32,530 --> 00:21:37,359 In doing this work with double angles, in effect, we've been 232 00:21:37,359 --> 00:21:41,749 looking at what are called multiple angles, and here I've 233 00:21:41,749 --> 00:21:46,578 been drawing sketches of a single angle. If you like, just 234 00:21:46,578 --> 00:21:52,724 sign X Cos X. So the question is, what does the graph of sine 235 00:21:52,724 --> 00:21:54,041 2X look like? 236 00:21:54,670 --> 00:22:00,429 What does the graph of cause 3X look like? Or for that matter, 237 00:22:00,429 --> 00:22:03,087 what about sign of 1/2 X? 238 00:22:03,100 --> 00:22:08,824 Well, let's just explore that sign X. Let's just have a look 239 00:22:08,824 --> 00:22:10,255 at its graph. 240 00:22:11,240 --> 00:22:14,795 Between North And 241 00:22:14,795 --> 00:22:21,140 2π So that's sine 242 00:22:21,140 --> 00:22:26,280 X. What about sign 2X? 243 00:22:28,740 --> 00:22:32,108 Over the same range. 244 00:22:33,330 --> 00:22:39,750 I just think about it. What are we doing when we're multiplying 245 00:22:39,750 --> 00:22:45,807 by two? Well, we're doubling yes, but what that means is 246 00:22:45,807 --> 00:22:48,122 everything that happens on this 247 00:22:48,122 --> 00:22:52,567 graph. Happened twice as fast on this graph. 248 00:22:53,920 --> 00:22:59,560 So in the time it takes this to go from North to 2π, it's done 249 00:22:59,560 --> 00:23:01,816 all of that in this space. 250 00:23:02,970 --> 00:23:10,374 So that bit of graph appears in this space so that we 251 00:23:10,374 --> 00:23:17,161 have up down there. Then of course it's periodic, so it 252 00:23:17,161 --> 00:23:19,012 does it again. 253 00:23:20,700 --> 00:23:24,360 So sign of 2 X. 254 00:23:26,220 --> 00:23:31,164 Gets through everything twice as quickly as sign X and sign of 255 00:23:31,164 --> 00:23:36,932 three. X will get through it 3 times as quickly, so I'll have 3 256 00:23:36,932 --> 00:23:41,464 copies of this graph in the same space, not to pie. 257 00:23:42,060 --> 00:23:48,720 And the same with cause 2X and cause 3X and cost 4X. 258 00:23:49,290 --> 00:23:54,519 What if I take sign of 1/2 of X? 259 00:23:54,520 --> 00:24:01,430 Do that. Sign of 1/2 of X. 260 00:24:02,210 --> 00:24:09,425 I'll do sign of X first just to give us the picture again. 261 00:24:09,940 --> 00:24:13,760 That's our graph between North 262 00:24:13,760 --> 00:24:17,269 and 2π. So what about? 263 00:24:17,840 --> 00:24:23,198 Sign. Of half of X on the same scale. 264 00:24:24,160 --> 00:24:27,385 Well, things are happening half 265 00:24:27,385 --> 00:24:34,310 as quickly. 1/2 of two pies just pie, so we will only 266 00:24:34,310 --> 00:24:40,834 have got through that bit of the curve by the time we've got to 267 00:24:40,834 --> 00:24:47,358 2π, so that in effect the graph of sign a half pie looks like 268 00:24:47,358 --> 00:24:49,222 that and continues on. 269 00:24:49,890 --> 00:24:52,450 In double the space. 270 00:24:53,900 --> 00:24:58,509 So graphs of multiple angles look a little bit different to 271 00:24:58,509 --> 00:24:59,766 the ordinary angle. 272 00:25:00,480 --> 00:25:05,492 But the thing that we have to remember is that if we have a 273 00:25:05,492 --> 00:25:06,924 multiple here, that's greater 274 00:25:06,924 --> 00:25:11,780 than one. Then it's going to get through it much quicker and 275 00:25:11,780 --> 00:25:13,730 we're going to see the graph 276 00:25:13,730 --> 00:25:18,262 repeated. If we've got a multiple here, that's less than 277 00:25:18,262 --> 00:25:21,934 one, it's going to take longer to get through. 278 00:25:22,810 --> 00:25:28,282 The graph and we're going to see the graph extended and drawn 279 00:25:28,282 --> 00:25:34,452 out. Let's have a look at the 280 00:25:34,452 --> 00:25:40,066 graph of cause X and cause 2X. 281 00:25:42,760 --> 00:25:45,448 Again between North. 282 00:25:45,950 --> 00:25:48,150 And. 2π 283 00:25:49,430 --> 00:25:51,430 So there's our graph. 284 00:25:53,980 --> 00:26:00,217 What about? Mark the points in the same positions. This is 285 00:26:00,217 --> 00:26:06,013 going to go through twice as quickly, so we're going to see 286 00:26:06,013 --> 00:26:11,326 that shape repeated in this space here, so we're going to 287 00:26:11,326 --> 00:26:18,088 see that come down and go up, and then we're going to see it 288 00:26:18,088 --> 00:26:21,895 repeated again. I thought so. Sketching these graphs of 289 00:26:21,895 --> 00:26:25,990 multiple angles is quite easy. All you need to do is look at 290 00:26:25,990 --> 00:26:29,455 the original graph and judge the number of times that it's 291 00:26:29,455 --> 00:26:31,975 going to be repeated over the given range.