[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.82,0:00:02.21,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:00:02.25,0:00:03.64,Default,,0000,0000,0000,,Let's take a look at how to solve
Dialogue: 0,0:00:03.64,0:00:06.32,Default,,0000,0000,0000,,some composition of trig functions with their inverses.
Dialogue: 0,0:00:06.53,0:00:09.52,Default,,0000,0000,0000,,So, all of these functions today are going to be
Dialogue: 0,0:00:09.96,0:00:12.37,Default,,0000,0000,0000,,a trig function inverse of a trig function of an angle.
Dialogue: 0,0:00:13.14,0:00:16.92,Default,,0000,0000,0000,,Often when you evaluate composition of functions in the inverses,
Dialogue: 0,0:00:16.94,0:00:17.66,Default,,0000,0000,0000,,it's very easy.
Dialogue: 0,0:00:17.70,0:00:20.33,Default,,0000,0000,0000,,You just take whatever that input is as your answer.
Dialogue: 0,0:00:20.54,0:00:23.01,Default,,0000,0000,0000,,That's not going to work here because we've restricted
Dialogue: 0,0:00:23.42,0:00:24.61,Default,,0000,0000,0000,,the range of
Dialogue: 0,0:00:24.77,0:00:26.49,Default,,0000,0000,0000,,the inverse trig functions so much.
Dialogue: 0,0:00:26.90,0:00:27.22,Default,,0000,0000,0000,,So,
Dialogue: 0,0:00:28.28,0:00:31.56,Default,,0000,0000,0000,,these first three examples we have the same angle 5 pi over 8.
Dialogue: 0,0:00:31.65,0:00:33.11,Default,,0000,0000,0000,,Let's see where that is on a unit circle.
Dialogue: 0,0:00:33.19,0:00:35.06,Default,,0000,0000,0000,,It's a good idea whenever you're doing these to
Dialogue: 0,0:00:35.39,0:00:36.37,Default,,0000,0000,0000,,draw a circle
Dialogue: 0,0:00:36.83,0:00:37.30,Default,,0000,0000,0000,,and
Dialogue: 0,0:00:37.55,0:00:38.15,Default,,0000,0000,0000,,draw in the angle.
Dialogue: 0,0:00:38.19,0:00:40.28,Default,,0000,0000,0000,,That's about approximately 5 pi over 8.
Dialogue: 0,0:00:40.75,0:00:45.67,Default,,0000,0000,0000,,We can't say that sine inverse of sine of 5 pi over 8 is just 5 pi over 8 because
Dialogue: 0,0:00:46.11,0:00:50.26,Default,,0000,0000,0000,,our answer to any sine inverse question needs to be in the 4th or 1st quadrant,
Dialogue: 0,0:00:50.59,0:00:54.35,Default,,0000,0000,0000,,specifically between negative pi over 2 and positive pi over 2.
Dialogue: 0,0:00:55.06,0:00:57.24,Default,,0000,0000,0000,,So, we need to figure out which angle
Dialogue: 0,0:00:57.81,0:01:00.79,Default,,0000,0000,0000,,has the same sine value as 5 pi over 8.
Dialogue: 0,0:01:01.33,0:01:03.40,Default,,0000,0000,0000,,So, if we just come straight across,
Dialogue: 0,0:01:03.77,0:01:05.36,Default,,0000,0000,0000,,reflect over the y-axis,
Dialogue: 0,0:01:05.57,0:01:07.61,Default,,0000,0000,0000,,that will give us a point that has the same y
Dialogue: 0,0:01:07.61,0:01:10.08,Default,,0000,0000,0000,,value. Place on the unit circle with the same y value.
Dialogue: 0,0:01:10.33,0:01:11.70,Default,,0000,0000,0000,,We just want to figure out
Dialogue: 0,0:01:11.97,0:01:13.41,Default,,0000,0000,0000,,what is this angle here.
Dialogue: 0,0:01:14.72,0:01:15.32,Default,,0000,0000,0000,,Well,
Dialogue: 0,0:01:15.61,0:01:17.76,Default,,0000,0000,0000,,to get to 5 pi over 8,
Dialogue: 0,0:01:17.89,0:01:19.65,Default,,0000,0000,0000,,we could have started on the negative
Dialogue: 0,0:01:20.38,0:01:22.74,Default,,0000,0000,0000,,x-axis and rotated backwards
Dialogue: 0,0:01:23.05,0:01:26.96,Default,,0000,0000,0000,,3/8 of pi because 5 is 3 from 8.
Dialogue: 0,0:01:27.27,0:01:30.32,Default,,0000,0000,0000,,So, let's just go forward 3 pi over 8
Dialogue: 0,0:01:30.75,0:01:34.12,Default,,0000,0000,0000,,from the positive x-axis so we get 3 pi
Dialogue: 0,0:01:35.50,0:01:36.02,Default,,0000,0000,0000,,over 8.
Dialogue: 0,0:01:37.35,0:01:39.38,Default,,0000,0000,0000,,Tangent inverse of tangent 5 pi over 8
Dialogue: 0,0:01:39.39,0:01:40.86,Default,,0000,0000,0000,,is not going to be the same thing.
Dialogue: 0,0:01:41.47,0:01:45.46,Default,,0000,0000,0000,,We need to find a place in the unit circle where y over x is the same
Dialogue: 0,0:01:45.75,0:01:46.98,Default,,0000,0000,0000,,as it is at
Dialogue: 0,0:01:47.27,0:01:49.99,Default,,0000,0000,0000,,this point up here where the angle is 5 pi over 8.
Dialogue: 0,0:01:51.24,0:01:54.23,Default,,0000,0000,0000,,We need therefore to be in the 4th quadrant because
Dialogue: 0,0:01:54.51,0:01:58.34,Default,,0000,0000,0000,,tangent is negative in the 2nd quadrant and also in the 4th quadrant.
Dialogue: 0,0:01:58.51,0:02:00.30,Default,,0000,0000,0000,,So, what we're going to do is we're going to actually rotate
Dialogue: 0,0:02:00.74,0:02:03.10,Default,,0000,0000,0000,,pi radians a 180 degrees around.
Dialogue: 0,0:02:03.61,0:02:04.06,Default,,0000,0000,0000,,So,
Dialogue: 0,0:02:04.51,0:02:07.90,Default,,0000,0000,0000,,this angle here must be negative
Dialogue: 0,0:02:08.43,0:02:10.27,Default,,0000,0000,0000,,3 pi over 8.
Dialogue: 0,0:02:10.31,0:02:11.98,Default,,0000,0000,0000,,I figured that out by again saying
Dialogue: 0,0:02:12.15,0:02:14.79,Default,,0000,0000,0000,,I'm 3 pi over 8 away from the negative x-axis,
Dialogue: 0,0:02:14.87,0:02:18.59,Default,,0000,0000,0000,,so I need a backup 3 pi over 8 from the positive x-axis.
Dialogue: 0,0:02:19.13,0:02:21.84,Default,,0000,0000,0000,,What about cosine inverse of cosine of 5 pi over 8?
Dialogue: 0,0:02:21.88,0:02:23.27,Default,,0000,0000,0000,,What do we need to change there?
Dialogue: 0,0:02:23.64,0:02:23.81,Default,,0000,0000,0000,,Well,
Dialogue: 0,0:02:24.04,0:02:24.92,Default,,0000,0000,0000,,turns out nothing.
Dialogue: 0,0:02:25.88,0:02:28.87,Default,,0000,0000,0000,,Unlike sine and tangent inverse,
Dialogue: 0,0:02:29.59,0:02:30.84,Default,,0000,0000,0000,,cosine inverse
Dialogue: 0,0:02:31.28,0:02:31.79,Default,,0000,0000,0000,,is
Dialogue: 0,0:02:32.60,0:02:36.06,Default,,0000,0000,0000,,always going to give us an answer in the first or second quadrant.
Dialogue: 0,0:02:36.24,0:02:38.03,Default,,0000,0000,0000,,5 pi over 8 is already in the second quadrant,
Dialogue: 0,0:02:38.07,0:02:39.87,Default,,0000,0000,0000,,so we don't need to change anything at all.
Dialogue: 0,0:02:41.60,0:02:42.34,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:02:42.54,0:02:43.68,Default,,0000,0000,0000,,What about
Dialogue: 0,0:02:44.30,0:02:48.19,Default,,0000,0000,0000,,this second angle 12 pi over 7?
Dialogue: 0,0:02:48.62,0:02:49.17,Default,,0000,0000,0000,,Let's draw that in.
Dialogue: 0,0:02:49.18,0:02:50.73,Default,,0000,0000,0000,,That's going to be in the 4th quadrant,
Dialogue: 0,0:02:51.02,0:02:52.25,Default,,0000,0000,0000,,maybe about right there.
Dialogue: 0,0:02:53.30,0:02:54.20,Default,,0000,0000,0000,,And
Dialogue: 0,0:02:54.90,0:02:57.43,Default,,0000,0000,0000,,inverse of sine of 12 pi over 7,
Dialogue: 0,0:02:57.48,0:03:01.51,Default,,0000,0000,0000,,you might think it's just 12 pi over 7 because we're already in the fourth quadrant.
Dialogue: 0,0:03:01.88,0:03:05.00,Default,,0000,0000,0000,,But we need to name the angle in such a way that the
Dialogue: 0,0:03:05.00,0:03:08.20,Default,,0000,0000,0000,,angle is between negative pi over 2 and positive pi over 2.
Dialogue: 0,0:03:08.28,0:03:10.60,Default,,0000,0000,0000,,So, we don't have to change the position on the unit circle.
Dialogue: 0,0:03:10.64,0:03:11.71,Default,,0000,0000,0000,,We're in the right place.
Dialogue: 0,0:03:12.04,0:03:14.75,Default,,0000,0000,0000,,We need to give it the right name, and the right name in this case
Dialogue: 0,0:03:15.04,0:03:18.76,Default,,0000,0000,0000,,is negative 2 pi over 7.
Dialogue: 0,0:03:19.12,0:03:20.50,Default,,0000,0000,0000,,I know that because
Dialogue: 0,0:03:20.82,0:03:23.46,Default,,0000,0000,0000,,12 pi over 7 is just 2/7
Dialogue: 0,0:03:23.98,0:03:24.97,Default,,0000,0000,0000,,away from being
Dialogue: 0,0:03:25.38,0:03:26.59,Default,,0000,0000,0000,,14 pi over 7,
Dialogue: 0,0:03:26.66,0:03:28.62,Default,,0000,0000,0000,,which would be 2 pi a complete revolution.
Dialogue: 0,0:03:29.86,0:03:32.85,Default,,0000,0000,0000,,How about tangent inverse of tangent of 12 pi over 7?
Dialogue: 0,0:03:33.13,0:03:33.25,Default,,0000,0000,0000,,Well,
Dialogue: 0,0:03:33.45,0:03:33.77,Default,,0000,0000,0000,,same deal.
Dialogue: 0,0:03:33.85,0:03:35.36,Default,,0000,0000,0000,,We're in the correct place.
Dialogue: 0,0:03:35.45,0:03:38.20,Default,,0000,0000,0000,,We don't need to rotate at all or reflect over the y-axis.
Dialogue: 0,0:03:38.53,0:03:40.52,Default,,0000,0000,0000,,So, we just need to give it the right name,
Dialogue: 0,0:03:40.57,0:03:42.65,Default,,0000,0000,0000,,which is -2 pi over 7.
Dialogue: 0,0:03:44.20,0:03:47.34,Default,,0000,0000,0000,,But what about cosine inverse of cosine 12 pi over 7?
Dialogue: 0,0:03:47.51,0:03:48.71,Default,,0000,0000,0000,,Here we do have to do something.
Dialogue: 0,0:03:48.75,0:03:49.90,Default,,0000,0000,0000,,We have to change the point
Dialogue: 0,0:03:50.43,0:03:52.82,Default,,0000,0000,0000,,because we're not in the first or second quadrant.
Dialogue: 0,0:03:53.15,0:03:54.30,Default,,0000,0000,0000,,How do we get to the right place?
Dialogue: 0,0:03:54.39,0:03:54.51,Default,,0000,0000,0000,,Well,
Dialogue: 0,0:03:54.67,0:03:57.94,Default,,0000,0000,0000,,we want to find a place in the unit circle that is the same x value.
Dialogue: 0,0:03:58.22,0:03:59.98,Default,,0000,0000,0000,,So, let's go straight on up,
Dialogue: 0,0:04:00.39,0:04:02.07,Default,,0000,0000,0000,,and it must be right around there.
Dialogue: 0,0:04:02.75,0:04:04.38,Default,,0000,0000,0000,,So, what is this angle here?
Dialogue: 0,0:04:05.16,0:04:06.78,Default,,0000,0000,0000,,That angle there,
Dialogue: 0,0:04:07.43,0:04:11.99,Default,,0000,0000,0000,,we went backwards 2 pi over 7 to get down to 12 pi over 7.
Dialogue: 0,0:04:12.03,0:04:13.34,Default,,0000,0000,0000,,So, let's go forwards
Dialogue: 0,0:04:13.51,0:04:14.42,Default,,0000,0000,0000,,the same amount,
Dialogue: 0,0:04:14.59,0:04:17.91,Default,,0000,0000,0000,,so this will be positive 2 pi over 7.
Dialogue: 0,0:04:19.18,0:04:19.45,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:04:19.53,0:04:20.56,Default,,0000,0000,0000,,So that's how you solve those.
Dialogue: 0,0:04:20.89,0:04:23.21,Default,,0000,0000,0000,,Why don't you try a couple on your own here?
Dialogue: 0,0:04:23.57,0:04:24.52,Default,,0000,0000,0000,,Let's move this over.
Dialogue: 0,0:04:24.69,0:04:26.48,Default,,0000,0000,0000,,I have a few for you to try out.
Dialogue: 0,0:04:26.69,0:04:27.12,Default,,0000,0000,0000,,Try those.
Dialogue: 0,0:04:27.17,0:04:27.74,Default,,0000,0000,0000,,Pause the video,
Dialogue: 0,0:04:27.85,0:04:28.33,Default,,0000,0000,0000,,try those,
Dialogue: 0,0:04:28.41,0:04:30.85,Default,,0000,0000,0000,,and then I'll come back and tell you if you have the right answer.
Dialogue: 0,0:04:35.43,0:04:36.07,Default,,0000,0000,0000,,All right,
Dialogue: 0,0:04:36.25,0:04:37.06,Default,,0000,0000,0000,,let's take a look.
Dialogue: 0,0:04:37.33,0:04:38.76,Default,,0000,0000,0000,,6 pi over 5.
Dialogue: 0,0:04:38.89,0:04:41.36,Default,,0000,0000,0000,,Good idea to quickly sketch a circle,
Dialogue: 0,0:04:41.65,0:04:43.84,Default,,0000,0000,0000,,unit circle and see where is that
Dialogue: 0,0:04:44.37,0:04:45.82,Default,,0000,0000,0000,,6 pi over 5.
Dialogue: 0,0:04:45.85,0:04:47.73,Default,,0000,0000,0000,,It's just over 5 pi over 5,
Dialogue: 0,0:04:47.77,0:04:48.61,Default,,0000,0000,0000,,which is half a circle.
Dialogue: 0,0:04:48.69,0:04:49.96,Default,,0000,0000,0000,,So, we're right around there.
Dialogue: 0,0:04:51.02,0:04:51.45,Default,,0000,0000,0000,,Now,
Dialogue: 0,0:04:51.62,0:04:52.67,Default,,0000,0000,0000,,we're in the 3rd quadrant,
Dialogue: 0,0:04:52.70,0:04:55.41,Default,,0000,0000,0000,,so we do need to change the position of the point
Dialogue: 0,0:04:55.94,0:04:56.49,Default,,0000,0000,0000,,for
Dialogue: 0,0:04:56.81,0:04:57.81,Default,,0000,0000,0000,,sine inverse.
Dialogue: 0,0:04:58.14,0:05:00.41,Default,,0000,0000,0000,,We want to reflect over the y-axis,
Dialogue: 0,0:05:00.53,0:05:01.58,Default,,0000,0000,0000,,so we'll have the same y value.
Dialogue: 0,0:05:01.66,0:05:02.69,Default,,0000,0000,0000,,So, we need to be right there.
Dialogue: 0,0:05:02.94,0:05:06.90,Default,,0000,0000,0000,,So, instead of going 1/5 of pi beyond the x-axis,
Dialogue: 0,0:05:06.98,0:05:08.69,Default,,0000,0000,0000,,we're going to come back a 5th of pi.
Dialogue: 0,0:05:09.06,0:05:10.53,Default,,0000,0000,0000,,This should be negative
Dialogue: 0,0:05:11.69,0:05:12.73,Default,,0000,0000,0000,,pi over 5.
Dialogue: 0,0:05:14.03,0:05:14.67,Default,,0000,0000,0000,,For tangent,
Dialogue: 0,0:05:14.68,0:05:15.70,Default,,0000,0000,0000,,we also need to change,
Dialogue: 0,0:05:15.77,0:05:18.07,Default,,0000,0000,0000,,but now we want to be in the first quadrant because the
Dialogue: 0,0:05:18.07,0:05:20.70,Default,,0000,0000,0000,,tangent will be positive like it is in the 3rd quadrant.
Dialogue: 0,0:05:21.02,0:05:24.95,Default,,0000,0000,0000,,So, in this case, it will be positive pi over 5.
Dialogue: 0,0:05:26.08,0:05:28.99,Default,,0000,0000,0000,,And for cosine inverse of cosine of 6 pi over 5,
Dialogue: 0,0:05:29.03,0:05:30.39,Default,,0000,0000,0000,,again we need to change the position because
Dialogue: 0,0:05:30.39,0:05:32.12,Default,,0000,0000,0000,,we're not in the first or second quadrant.
Dialogue: 0,0:05:32.43,0:05:35.63,Default,,0000,0000,0000,,So, we'll reflect over the x-axis that'll keep.
Dialogue: 0,0:05:36.76,0:05:38.48,Default,,0000,0000,0000,,It'll keep the x value the same.
Dialogue: 0,0:05:38.75,0:05:43.30,Default,,0000,0000,0000,,And so, now instead of going 1 more fifth of pi from the negative x-axis,
Dialogue: 0,0:05:43.35,0:05:44.51,Default,,0000,0000,0000,,we need a backup one.
Dialogue: 0,0:05:44.71,0:05:46.47,Default,,0000,0000,0000,,So, instead of 5 pi we go back one,
Dialogue: 0,0:05:46.48,0:05:47.10,Default,,0000,0000,0000,,that would be
Dialogue: 0,0:05:47.67,0:05:48.34,Default,,0000,0000,0000,,4
Dialogue: 0,0:05:48.79,0:05:50.07,Default,,0000,0000,0000,,pi over 5.
Dialogue: 0,0:05:50.99,0:05:51.52,Default,,0000,0000,0000,,All right,
Dialogue: 0,0:05:51.66,0:05:53.24,Default,,0000,0000,0000,,that's how you solve problems like this.
Dialogue: 0,0:05:53.41,0:05:54.43,Default,,0000,0000,0000,,I hope this has helped,
Dialogue: 0,0:05:54.45,0:05:55.45,Default,,0000,0000,0000,,and thanks for watching.