1 00:00:00,390 --> 00:00:03,490 In the last video, I showed you that if someone were to walk 2 00:00:03,490 --> 00:00:10,010 up to you and ask you what is the arcsine-- Whoops. 3 00:00:10,010 --> 00:00:13,190 --arcsine of x? 4 00:00:13,190 --> 00:00:16,360 And so this is going to be equal to who knows what. 5 00:00:16,360 --> 00:00:20,100 This is just the same thing as saying that the sine of 6 00:00:20,100 --> 00:00:22,470 some angle is equal to x. 7 00:00:22,470 --> 00:00:25,520 And we solved it in a couple of cases in the last example. 8 00:00:25,520 --> 00:00:28,230 So using the same pattern-- Let me show you this. 9 00:00:28,230 --> 00:00:31,810 I could have also rewritten this as the inverse sine 10 00:00:31,810 --> 00:00:33,540 of x is equal to what. 11 00:00:33,540 --> 00:00:35,140 These are equivalent statements. 12 00:00:35,140 --> 00:00:37,480 Two ways of writing the inverse sine function. 13 00:00:37,480 --> 00:00:39,790 This is more-- This is the inverse sine function. 14 00:00:39,790 --> 00:00:41,460 You're not taking this to the negative 1 power. 15 00:00:41,460 --> 00:00:45,340 You're just saying the sine of what-- So what question mark-- 16 00:00:45,340 --> 00:00:46,920 What angle is equal to x? 17 00:00:46,920 --> 00:00:48,470 And we did this in the last video. 18 00:00:48,470 --> 00:00:52,020 So by the same pattern, if I were to walk up to you on the 19 00:00:52,020 --> 00:00:58,770 street and I were to say the tangent of-- the inverse 20 00:00:58,770 --> 00:01:02,010 tangent of x is equal to what? 21 00:01:02,010 --> 00:01:05,140 You should immediately in your head say, oh he's just asking 22 00:01:05,140 --> 00:01:09,080 me-- He's just saying the tangent of some angle 23 00:01:09,080 --> 00:01:10,440 is equal to x. 24 00:01:10,440 --> 00:01:13,380 And I just need to figure out what that angle is. 25 00:01:13,380 --> 00:01:14,840 So let's do an example. 26 00:01:14,840 --> 00:01:17,060 So let's say I were walk up to you on the street. 27 00:01:17,060 --> 00:01:20,150 There's a lot of a walking up on a lot of streets. 28 00:01:20,150 --> 00:01:23,790 I would write -- And I were to say you what is the 29 00:01:23,790 --> 00:01:27,970 arctangent of minus 1? 30 00:01:27,970 --> 00:01:30,190 Or I could have equivalently asked you, what is the 31 00:01:30,190 --> 00:01:32,510 inverse tangent of minus 1? 32 00:01:32,510 --> 00:01:35,190 These are equivalent questions. 33 00:01:35,190 --> 00:01:37,430 And what you should do is you should, in your head-- If you 34 00:01:37,430 --> 00:01:40,200 don't have this memorized, you should draw the unit circle. 35 00:01:40,200 --> 00:01:42,640 Actually let me just do a refresher of what tangent 36 00:01:42,640 --> 00:01:44,330 is even asking us. 37 00:01:44,330 --> 00:01:48,890 The tangent of theta-- this is just the straight-up, vanilla, 38 00:01:48,890 --> 00:01:52,630 non-inverse function tangent --that's equal to the sine of 39 00:01:52,630 --> 00:01:56,690 theta over the cosine of theta. 40 00:01:56,690 --> 00:02:00,670 And the sine of theta is the y-value on the unit function-- 41 00:02:00,670 --> 00:02:03,010 on the unit circle. 42 00:02:03,010 --> 00:02:06,730 And the cosine of theta is the x-value. 43 00:02:06,730 --> 00:02:08,600 And so if you draw a line-- Let me draw a little 44 00:02:08,600 --> 00:02:11,110 unit circle here. 45 00:02:11,110 --> 00:02:14,770 So if I have a unit circle like that. 46 00:02:14,770 --> 00:02:17,980 And let's say I'm at some angle. 47 00:02:17,980 --> 00:02:20,940 Let's say that's my angle theta. 48 00:02:20,940 --> 00:02:25,640 And this is my y-- my coordinates x, y. 49 00:02:25,640 --> 00:02:29,380 We know already that the y-value, this is 50 00:02:29,380 --> 00:02:30,880 the sine of theta. 51 00:02:30,880 --> 00:02:32,780 Let me scroll over here. 52 00:02:32,780 --> 00:02:34,210 Sine of theta. 53 00:02:34,210 --> 00:02:38,730 And we already know that this x-value is the cosine of theta. 54 00:02:38,730 --> 00:02:40,200 So what's the tangent going to be? 55 00:02:40,200 --> 00:02:46,670 It's going to be this distance divided by this distance. 56 00:02:46,670 --> 00:02:49,970 Or from your algebra I, this might ring a bell, because 57 00:02:49,970 --> 00:02:52,520 we're starting at the origin from the point 0, 0. 58 00:02:52,520 --> 00:02:56,250 This is our change in y over our change in x. 59 00:02:56,250 --> 00:02:58,700 Or it's our rise over run. 60 00:02:58,700 --> 00:03:01,950 Or you can kind of view the tangent of theta, or it really 61 00:03:01,950 --> 00:03:04,570 is, as the slope of this line. 62 00:03:04,570 --> 00:03:05,730 The slope. 63 00:03:05,730 --> 00:03:11,660 So you could write slope is equal to the tangent of theta. 64 00:03:11,660 --> 00:03:14,350 So let's just bear that in mind when we go to our example. 65 00:03:14,350 --> 00:03:19,550 If I'm asking you-- and I'll rewrite it here --what is the 66 00:03:19,550 --> 00:03:22,600 inverse tangent of minus 1? 67 00:03:22,600 --> 00:03:23,880 And I'll keep rewriting it. 68 00:03:23,880 --> 00:03:26,440 Or the arctangent of minus 1? 69 00:03:26,440 --> 00:03:29,830 I'm saying what angle gives me a slope of minus 1 70 00:03:29,830 --> 00:03:31,320 on the unit circle? 71 00:03:31,320 --> 00:03:34,830 So let's draw the unit circle. 72 00:03:34,830 --> 00:03:37,960 Let's draw the unit circle like that. 73 00:03:37,960 --> 00:03:42,880 Then I have my axes like that. 74 00:03:42,880 --> 00:03:44,440 And I want a slope of minus 1. 75 00:03:44,440 --> 00:03:46,450 A slope of minus 1 looks like this. 76 00:03:49,995 --> 00:03:52,430 If it was like that, it would be slope of plus 1. 77 00:03:52,430 --> 00:03:55,580 So what angle is this? 78 00:03:55,580 --> 00:03:58,710 So in order to have a slope of minus 1, this distance is 79 00:03:58,710 --> 00:04:00,580 the same as this distance. 80 00:04:00,580 --> 00:04:03,940 And you might already recognize that this is a right angle. 81 00:04:03,940 --> 00:04:06,410 So these angles have to be the same. 82 00:04:06,410 --> 00:04:09,250 So this has to be a 45 45 90 triangle. 83 00:04:09,250 --> 00:04:10,630 This is an isosceles triangle. 84 00:04:10,630 --> 00:04:12,880 These two have to add up to 90 and they have to be the same. 85 00:04:12,880 --> 00:04:15,120 So this is 45 45 90. 86 00:04:15,120 --> 00:04:18,680 And if you know your 45 45 90-- Actually, you don't even have 87 00:04:18,680 --> 00:04:20,250 to know the sides of it. 88 00:04:20,250 --> 00:04:22,440 In the previous video, we saw that this is going 89 00:04:22,440 --> 00:04:23,810 to be-- Right here. 90 00:04:23,810 --> 00:04:28,040 This distance is going to be square root of 2 over 2. 91 00:04:28,040 --> 00:04:31,610 So this coordinate in the y-direction is minus 92 00:04:31,610 --> 00:04:33,380 square root of 2 over 2. 93 00:04:33,380 --> 00:04:36,210 And then this coordinate right here on the x-direction is 94 00:04:36,210 --> 00:04:39,500 square root of 2 over 2 because this length right 95 00:04:39,500 --> 00:04:40,960 there is that. 96 00:04:40,960 --> 00:04:43,430 So the square root of 2 over 2 squared plus the square root of 97 00:04:43,430 --> 00:04:46,170 2 over 2 squared is equal to 1 squared. 98 00:04:46,170 --> 00:04:47,806 But the important thing to realize is this is 99 00:04:47,806 --> 00:04:50,690 a 45 45 90 triangle. 100 00:04:50,690 --> 00:04:54,700 So this angle right here is-- Well if you're just looking at 101 00:04:54,700 --> 00:04:57,670 the triangle by itself, you would say that this is 102 00:04:57,670 --> 00:04:59,360 a 45 degree angle. 103 00:04:59,360 --> 00:05:04,060 But since we're going clockwise below the x-axis, we'll call 104 00:05:04,060 --> 00:05:05,980 this a minus 45 degree angle. 105 00:05:09,220 --> 00:05:13,710 So the tangent of minus 40-- Let me write that down. 106 00:05:13,710 --> 00:05:15,250 So if I'm in degrees. 107 00:05:15,250 --> 00:05:16,910 And that tends to be how I think. 108 00:05:16,910 --> 00:05:25,160 So I could write the tangent of minus 45 degrees it equals this 109 00:05:25,160 --> 00:05:28,167 negative value-- minus square root of 2 over 2 over square 110 00:05:28,167 --> 00:05:31,200 root of 2 over 2, which is equal to minus 1. 111 00:05:31,200 --> 00:05:36,670 Or I could write the arctangent of minus 1 is equal 112 00:05:36,670 --> 00:05:39,110 to minus 45 degrees. 113 00:05:39,110 --> 00:05:40,920 Now if we're dealing with radians, we just have to 114 00:05:40,920 --> 00:05:42,350 convert this to radians. 115 00:05:42,350 --> 00:05:47,530 So we multiply that times-- We get pi radians for 116 00:05:47,530 --> 00:05:49,880 every 180 degrees. 117 00:05:49,880 --> 00:05:51,890 The degrees cancel out. 118 00:05:51,890 --> 00:05:53,960 So you have a 45 over 180. 119 00:05:53,960 --> 00:05:55,160 This goes four times. 120 00:05:55,160 --> 00:05:57,570 So this is equal to-- you have the minus sign-- 121 00:05:57,570 --> 00:06:01,710 minus pi over 4 radians. 122 00:06:01,710 --> 00:06:06,450 So the arctangent of minus 1 is equal to minus pi over 4 or the 123 00:06:06,450 --> 00:06:13,850 inverse tangent of minus 1 is also equal to minus pi over 4. 124 00:06:13,850 --> 00:06:15,350 Now you could say, look. 125 00:06:15,350 --> 00:06:17,930 If I'm at minus pi over 4, that's there. 126 00:06:17,930 --> 00:06:18,540 That's fine. 127 00:06:18,540 --> 00:06:22,360 This gives me a value of minus 1 because the slope 128 00:06:22,360 --> 00:06:23,320 of this line is minus 1. 129 00:06:23,320 --> 00:06:25,120 But I can keep going around the unit circle. 130 00:06:25,120 --> 00:06:26,880 I could add 2 pi to this. 131 00:06:26,880 --> 00:06:30,890 Maybe I could add 2 pi to this and that would also give me-- 132 00:06:30,890 --> 00:06:33,090 If I took the tangent of that angle, it would also 133 00:06:33,090 --> 00:06:34,640 give me minus 1. 134 00:06:34,640 --> 00:06:39,170 Or I could add 2 pi again and it'll, again, give me minus 1. 135 00:06:39,170 --> 00:06:42,100 In fact I could go to this point right here. 136 00:06:42,100 --> 00:06:44,420 And the tangent would also give me minus 1 because 137 00:06:44,420 --> 00:06:45,790 the slope is right there. 138 00:06:45,790 --> 00:06:49,460 And like I said in the sine-- in the inverse sine video, you 139 00:06:49,460 --> 00:06:51,960 can't have a function that has a 1 to many mapping. 140 00:06:51,960 --> 00:06:58,190 You can't-- Tangent inverse of x can't map to a bunch 141 00:06:58,190 --> 00:06:59,830 of different values. 142 00:06:59,830 --> 00:07:03,280 It can't map to minus pi over 4. 143 00:07:03,280 --> 00:07:09,270 It can't map to 3-- what it would be? --3 pi over 4. 144 00:07:09,270 --> 00:07:09,730 I don't know. 145 00:07:09,730 --> 00:07:14,310 It would be-- I'll just say 2 pi minus pi over 4. 146 00:07:14,310 --> 00:07:16,200 Or 4 pi minus pi. 147 00:07:16,200 --> 00:07:18,550 It can't map to all of these different things. 148 00:07:18,550 --> 00:07:20,740 So I have to constrict the range on the 149 00:07:20,740 --> 00:07:22,270 inverse tan function. 150 00:07:22,270 --> 00:07:25,570 And we'll restrict it very similarly to the way we 151 00:07:25,570 --> 00:07:29,100 restricted the sine-- the inverse sine range. 152 00:07:29,100 --> 00:07:32,510 We're going to restrict it to the first and fourth quadrants. 153 00:07:32,510 --> 00:07:36,010 So the answer to your inverse tangent is always going to be 154 00:07:36,010 --> 00:07:37,470 something in these quadrants. 155 00:07:37,470 --> 00:07:39,950 But it can't be this point and that point. 156 00:07:39,950 --> 00:07:44,810 Because a tangent function becomes undefined at pi over 157 00:07:44,810 --> 00:07:46,350 2 and at minus pi ever 2. 158 00:07:46,350 --> 00:07:48,110 Because your slope goes vertical. 159 00:07:48,110 --> 00:07:50,260 You start dividing-- Your change in x is 0. 160 00:07:50,260 --> 00:07:52,890 You're dividing-- Your cosine of theta goes to 0. 161 00:07:52,890 --> 00:07:55,560 So if you divide by that, it's undefined. 162 00:07:55,560 --> 00:07:59,730 So your range-- So if I-- Let me write this down. 163 00:07:59,730 --> 00:08:03,430 So if I have an inverse tangent of x, I'm going to-- Well, 164 00:08:03,430 --> 00:08:06,330 what are all the values that the tangent can take on? 165 00:08:06,330 --> 00:08:11,800 So if I have the tangent of theta is equal to x, what are 166 00:08:11,800 --> 00:08:13,810 all the different values that x could take on? 167 00:08:13,810 --> 00:08:16,900 These are all the possible values for the slope. 168 00:08:16,900 --> 00:08:18,810 And that slope can take on anything. 169 00:08:18,810 --> 00:08:22,700 So x could be anywhere between minus infinity 170 00:08:22,700 --> 00:08:24,870 and positive infinity. 171 00:08:24,870 --> 00:08:27,250 x could pretty much take on any value. 172 00:08:27,250 --> 00:08:29,050 But what about theta? 173 00:08:29,050 --> 00:08:29,950 Well I just said it. 174 00:08:29,950 --> 00:08:33,910 Theta, you can only go from minus pi over 2 all 175 00:08:33,910 --> 00:08:35,260 the way to pi over 2. 176 00:08:35,260 --> 00:08:37,680 And you can't even include pi over 2 or minus pi over 2 177 00:08:37,680 --> 00:08:39,790 because then you'd be vertical. 178 00:08:39,790 --> 00:08:41,910 So then you say your-- So if I'm just dealing 179 00:08:41,910 --> 00:08:43,310 with vanilla tangent. 180 00:08:43,310 --> 00:08:44,470 Not the inverse. 181 00:08:44,470 --> 00:08:50,940 The domain-- Well the domain of tangent can go multiple times 182 00:08:50,940 --> 00:08:53,410 around, so let me not make that statement. 183 00:08:53,410 --> 00:08:55,580 But if I want to do inverse tangent so I don't have 184 00:08:55,580 --> 00:08:56,610 a 1 to many mapping. 185 00:08:56,610 --> 00:08:58,870 I want to cross out all of these. 186 00:08:58,870 --> 00:09:04,180 I'm going to restrict theta, or my range, to be greater than 187 00:09:04,180 --> 00:09:10,160 the minus pi over 2 and less than positive pi over 2. 188 00:09:10,160 --> 00:09:13,710 And so if I restrict my range to this right here and I 189 00:09:13,710 --> 00:09:16,190 exclude that point and that point. 190 00:09:16,190 --> 00:09:17,720 Then I can only get one answer. 191 00:09:17,720 --> 00:09:21,670 When I say tangent of what gives me a slope of minus 1? 192 00:09:21,670 --> 00:09:24,360 And that's the question I'm asking right there. 193 00:09:24,360 --> 00:09:25,320 There's only one answer. 194 00:09:25,320 --> 00:09:27,320 Because if I keep-- This one falls out of it. 195 00:09:27,320 --> 00:09:29,440 And obviously as I go around and around, those fall out of 196 00:09:29,440 --> 00:09:34,680 that valid range for theta that I was giving you. 197 00:09:34,680 --> 00:09:38,270 And then just to kind of make sure we did it right. 198 00:09:38,270 --> 00:09:39,700 Our answer was pi over 4. 199 00:09:39,700 --> 00:09:42,430 Let's see if we get that when we use our calculator. 200 00:09:42,430 --> 00:09:50,340 So the inverse tangent of minus 1 is equal to that. 201 00:09:50,340 --> 00:09:53,460 Let's see if that's the same thing as minus pi over 4. 202 00:09:53,460 --> 00:09:57,760 Minus pi over 4 is equal to that. 203 00:09:57,760 --> 00:09:59,100 So it is minus pi over 4. 204 00:09:59,100 --> 00:10:02,280 But it was good that we solved it without a calculator because 205 00:10:02,280 --> 00:10:06,160 it's hard to recognize this as minus pi over 4.