11 - Learn ArcSin, ArcCos & ArcTan (Inverse Sin, Cos & Tan) - Part 1 - Free Educational videos for Students in K-12

11 - Learn ArcSin, ArcCos & ArcTan (Inverse Sin, Cos & Tan) - Part 1 - By Math and Science

Transcript


00:00	Well , welcome back . The title of this lesson
00:02	is called Arc Sine . Our co sign Arc tangent
00:05	. This is part one . In other words ,
00:07	title of this lesson is inverse . Sine inverse Cosine
00:11	inverse tangent . So we have kind of a a
00:14	cousin to each of the trig functions and that would
00:17	be the inverse , the inverse of the sign ,
00:19	the inverse of the co sign , the inverse of
00:20	the tangent . Now , I'm really excited to teach
00:22	this lesson because Arc Sine Arcos in our candidate gets
00:25	very , very confusing with almost every book I've ever
00:28	seen . It's really not the fault of the book
00:31	. It's just that it's actually really hard to draw
00:33	on paper what you know , what's going on here
00:36	and get it across without actually talking to somebody .
00:39	So what I'm gonna do , I think in the
00:40	beginning is I've drawn some things myself and I want
00:43	to show you the punch line of what's actually happening
00:46	without any real explanation . Then I'm gonna walk you
00:49	through a detailed , uh I don't want to call
00:52	it a derivation but a detailed sequence of of of
00:54	problems that shows you exactly why the ark assigned the
00:58	arc sine . The arc tangent behave the way that
01:00	they do because it is not obvious the first time
01:02	you look at it . But I can tell you
01:04	for those of you who think well , I'll just
01:06	blow through this . No problem . It's just the
01:08	opposite of the co signer , opposite of the sign
01:10	. I really encourage you to watch the entire lesson
01:12	and practice the problems because there are several gouaches in
01:15	here that will come back to haunt you in the
01:17	future . I can say that you will use this
01:19	stuff all throughout algebra , all throughout trig and pre
01:23	calculus , all throughout calculus and definitely all throughout Physics
01:26	and engineering . So take a moment here and listen
01:29	to what we're talking about . So you understand exactly
01:31	how these things behave . Okay , up until now
01:35	we have been taking the sign , the co signed
01:37	the tangent whatever of an angle . So for instance
01:40	, sign of 30 is one half , which is
01:42	the same thing as sine of pi over six radiance
01:45	is one half because five or six radiance is 30
01:48	degrees . Okay , so in your mind think Sine
01:50	of pi over six is one half . Sine of
01:52	30 is one half . That's the direction we've been
01:54	going . So there is an opposite operation to all
01:57	of these functions called the inverse . Remember the inverse
02:00	of a function just does the opposite . It kind
02:02	of undoes the original function . Right ? So basically
02:05	the arc sine does the opposite operation of the sign
02:08	. So for instance , if you know , and
02:10	I'm gonna write all this on the board , so
02:11	don't stress out here in the beginning , but verbally
02:13	just follow me here . If you know that the
02:15	sine of pi over six is one half , then
02:18	you know that the arc sine of one half is
02:22	giving you back an angle of pi over six .
02:25	Let me say that one more time when you do
02:26	the arc sine operation of the number , you're asking
02:29	for the angle back . So instead of taking the
02:32	sine of an angle to get a number , we're
02:34	gonna take the arc sine of some number and get
02:37	the angle back . So it's the way that we
02:39	find the angles , how we reverse , solve for
02:42	the angles in a triangle or an equation or whatever
02:44	is going to be basically the arc sine , the
02:46	arcos in the arc tangent , it goes the opposite
02:48	direction . So we're going to be taking the ark
02:51	assigned the arc tangent , the dark side of some
02:53	number and getting angle back from it . However ,
02:56	there's a huge gotcha here and that is the way
02:58	the unit circle works . There's always multiple angles around
03:01	the unit circle that can give you the same sign
03:05	. So there's always gonna be multiple choices . And
03:07	so we have uh in your books , you'll see
03:09	exactly how we lock it down and and to know
03:12	what angles the calculator the computer is going to give
03:15	back to you . So we have to talk through
03:17	that . So again , let me give you the
03:19	punchline . Then we're gonna go through a quite a
03:21	detailed set of problems so that you understand exactly where
03:24	it's coming from here is the punchline for those of
03:26	you who want to know the punch line . I
03:28	am not gonna go through all of this right now
03:30	because I'm going to go through it in detail in
03:32	the lesson . But we have this function called the
03:35	inverse sine . Also called the ark sign . It
03:37	is the opposite operation of a sign . You feed
03:40	numbers into the arc sine operation and you get angles
03:44	back . The numbers that you feed into the arc
03:47	sine can only be from plus one to minus one
03:49	. I'll explain why later you put those numbers into
03:52	the arc sine . You press the button on your
03:54	calculator , outcomes and angle . But the calculator will
03:57	only give you angles from negative pi over 22 pi
03:59	over to these angles over here . If you hit
04:02	any inverse sine of anything on the calculator when it
04:05	gives you an answer back , it will never be
04:06	outside of this range , negative 90 to 90 degrees
04:09	or negative pi over 22 pi over two radiance .
04:12	I will explain exactly why . As we go through
04:14	the lecture here , then we have an inverse cosine
04:17	operation called also called arc Co sign . Because the
04:21	co sign can only be between negative one and one
04:24	. I'll explain why in a minute . Then the
04:26	input to the art co sign function can only be
04:29	between negative one and positive one . And because of
04:32	that , the angles that the calculator will give back
04:34	as the baseline angles can only fall between zero and
04:38	pi . You see negative one and one fall on
04:40	the X axis here . So the angles that you
04:42	will get back will only be up here in the
04:43	top half of the unit circle zero to pi or
04:46	0 to 180 . That's what this means . But
04:49	greater than zero . Less than pi . Greater than
04:51	zero . Less than 1 80 . If you stick
04:54	a number in your calculator and our co sign ,
04:56	inverse co sign . You will never get an angle
04:58	outside of that range . Okay , arc tangent is
05:01	the one that's really bizarre and weird and hard to
05:03	understand it . First there's an inverse tangent called arc
05:06	tangent operation . The inputs to the arc tangent can
05:09	be way outside of the range of negative one ,
05:11	positive one . They can be negative infinity . Up
05:13	to positive infinity . I will explain as we go
05:15	through the lesson why that's the case . The numbers
05:18	that you can feed into this operator . This is
05:20	like a function essentially here that you can put into
05:23	there can be any number at all , from negative
05:25	infinity to positive infinity . But the angles that you
05:27	get out of the arc tangent operation will only be
05:30	between negative pi over two and pi over two negative
05:34	pi over 22 pi over two or negative 90 deposited
05:36	90 . So this is the punch line . I
05:38	don't expect you to understand anything other than the fact
05:42	that when you do an arc sine operation or you
05:44	do an art co sign operation or you do an
05:47	arc tangent operation . What you're getting out as an
05:49	answer is an angle and the angle that you get
05:52	back out of each one of those operations has limits
05:55	to it . Your calculator or computer will never give
05:58	you an angle outside of those limits . For an
06:00	arc sine , you will always get an angle between
06:03	negative pi over two or pi over two back which
06:05	is negative 90 deposited 90 For anarcho sign , you
06:08	will always get an angle between zero and pi ,
06:10	which means zero and 100 and 80 for an arc
06:12	tangent . You will always get an angle between negative
06:15	pi over 22 pi over two Which is negative 92
06:18	positive 90°. . That's the punchline . Now , the
06:21	question is why is that ? We have to dive
06:24	into it . I don't want you to just memorize
06:26	things . I want you to understand things . So
06:28	, let's cover this up . We will revisit this
06:30	in great detail . I promise you . And let's
06:32	go down and talk exactly through the logic of why
06:35	all this works . It is more then a one
06:38	or a two minute thing . I can't compress it
06:40	into three minutes . So please watch the whole thing
06:42	and you will understand exactly the reasons why . All
06:45	right . So , we know we know for instance
06:47	that the sign of pi over six , which is
06:51	30 degrees right ? Remember five or six or 30
06:53	degrees is what you should all remember . Sign of
06:56	30 is the famous one half . I've been telling
06:58	you sign of 30 is one half . Sine of
06:59	30 is one half . Now that we know radiant
07:02	sine of pi over six is one half . So
07:04	how do we go backwards ? If we know the
07:06	sine of pi over six is one half , then
07:09	we should be able to say the following thing .
07:12	Arc sine which is inverse sine or the opposite operation
07:16	of the sign of whatever the kind of the answer
07:20	is one half . Let me do it this way
07:22	, let me do it this way arc sine of
07:25	um yeah , let me do it this way .
07:27	Arc sine of one half is going to equal what
07:31	angle like this is what this is basically asking you
07:33	to do Arc sine of one half is what angle
07:36	. Another way to write this , You can kind
07:40	of translate this problem . I mean we've written it
07:41	down here but it's the same thing as saying the
07:43	sign of some angle is equal to one half .
07:47	When you say arc sine of one half . What
07:49	you're really asking is a sign of what unknown angle
07:52	is equal to one half . And you know in
07:54	your mind that sign of 30 is one half .
07:56	So it makes sense . Right sign of 30 is
07:58	1/2 30°. . So we can then say that this
08:01	angle is 30°, , but we don't like to talk
08:03	about degrees too much now . So we say it's
08:05	actually five or six radiance . Right ? So one
08:08	way to write it is called arc sine . That's
08:11	the cleanest way to write it . That means inverse
08:14	of the sign operation going backwards . Give me the
08:17	number , I give you the angle , that's what
08:19	it is , right . But you can also write
08:21	this in the following way . You can write it
08:24	in a different way and sometimes you'll see it written
08:26	like this in different books . You might see it
08:28	written as the following sign with a little negative one
08:31	up here of one half Is equal to Pi over
08:36	six . This representation is exactly the same as this
08:40	representation . So when you see sign to the negative
08:43	one power , it does not mean that it's raised
08:46	to the negative one power . It doesn't okay ,
08:48	it does not mean one over sine . Okay .
08:52	It doesn't mean that the negative one is not an
08:55	exponent . The negative one is if you remember back
08:58	we talked about inverse functions a long time ago we
09:01	said that negative one up there can mean inverse function
09:04	. So this literally means inverse function of the sine
09:07	function . So this is the inverse sine of some
09:10	number , gives me some angle again . Going backwards
09:13	. This is very confusing to write down because the
09:15	negative one looks like an exponent . So it confuses
09:18	a lot of people so it's much more clean to
09:20	say arc sine , but you can write it either
09:23	way and you'll see it either way in lots of
09:25	books , you'll see in calculus or whatever . You'll
09:27	see it written both ways , depending on the author
09:30	. Okay , so it returns an angle back .
09:33	Let's give another quick example and then I'm gonna show
09:36	you the huge gotcha in here about why it's so
09:39	confusing in just a second . So let's go and
09:40	take a look at another one that we know very
09:42	, very easily are very , very rapidly . What
09:45	is the ark co sign of square root of 3/2
09:52	? So we want to figure out we have the
09:54	inverse of the cosine function . That means we want
09:57	to take as an input , not an angle ,
09:59	we want to take , kind of what the unit
10:01	circle gives you around the outside and red the number
10:04	coming off the unit circle . And I want to
10:05	get the angle back . That's what the arc co
10:08	sign the arc sine does for . So what you
10:10	can do in order to kind of translate this in
10:13	your mind is you can then say that this is
10:15	the same thing as saying that the co sign of
10:18	some unknown angle is the square of 3/2 and the
10:21	unknown angle is what you're kind of getting here .
10:23	So the co sign of what angle here is going
10:26	to do that . The coastline of pi over six
10:29	degrees is a square to 3/2 , which means that
10:33	kind of coming up in translating appear the arco sign
10:35	of square to 3/2 is the angle pi over six
10:38	or 30 degrees . How do we know that ?
10:40	Because we know the sign of 30 degrees is one
10:42	half the co sign of 30 degrees is the other
10:45	number square root of 3/2 . So the co sign
10:47	of 30 degrees square to 3/2 , which means the
10:49	art co sign of the answer is the angle coming
10:52	back . Yeah . Okay . And of course ,
10:55	another way to write this that you might see in
10:58	other books down the road . Another way to write
11:00	this is the co sign inverse . This means inverse
11:04	. Cosine does not mean one over co sign or
11:06	co signs of the negative . One power of the
11:09	number square root of 3/2 is the angle coming back
11:14	, pi over six . Okay , pi over six
11:20	. So that's another example . Let's do another quick
11:24	example here that we can talk about . What about
11:27	arc tangent of square root of three ? Arc tangent
11:33	of square root of three is equal to what another
11:35	way to do this is to translate it in your
11:37	mind when you see arc tangent of a number of
11:39	the way you do it in your mind . Or
11:40	on your paper Actually as you say , the tangent
11:44	of some unknown angle that I don't know is going
11:47	to be equal to the square root of three .
11:49	Okay , now the angle that actually works here is
11:52	pi over pi over three . So the answer to
11:57	the arc tangent of this going backwards is the angle
11:59	pi over three . Right ? How do I know
12:02	it's pi over three . Well , I I know
12:04	the answer but the reason is because the tangent is
12:06	the same as the sign . So put sine of
12:09	pi over three over co sign of Pi over three
12:16	. What is the sine of pi over three ?
12:18	That's 60 degrees right . Sine of pi over three
12:20	is 60 degrees . So you no sign of 60
12:22	degrees . Is the square root of 3/2 . And
12:25	then the co sign of 60 degrees , you know
12:28	, is one half . If you flip and multiply
12:30	the twos will cancel and what you get back out
12:32	of it is the square root of three . So
12:34	, you have to bust this tangent up into a
12:36	sign and a co sign to figure it out .
12:37	Unless you have it memorized . I don't actually have
12:39	them memorized . So I have to go figure it
12:41	out . That the square root of three is the
12:43	is the result when you take the tangent of pi
12:45	over three . So , because of that , the
12:47	arc tangent of the pi over three squared of three
12:50	. Uh Number is the angle pi over three .
12:54	All right . And you can also write this in
12:57	another way you can write this as or uh tangent
13:02	Inverse of Square root of three is high over three
13:08	, however , three . All right . So arc
13:11	tangent , square root of three is the same thing
13:13	as inverse tangent written with the negative one of square
13:16	root of three . Do not write this or think
13:19	that the -1 is an exponent . Do not try
13:21	to add exponents . Do not try to subtract exponents
13:24	don't raise the exponent to a square root and cancel
13:27	them . Don't do any of that stuff because it's
13:29	not a real exponent . It's just a notation that
13:31	means arc tangent or arc sine or cosine . That's
13:34	what it means . Now , the last thing I
13:36	want to leave you with before we go into the
13:38	details of why this is kind of so confusing is
13:42	the following thing . You know . A long time
13:44	ago I introduced the concept of a function as a
13:47	box . A function is a box . You stick
13:50	numbers coming into this box . Inside the box is
13:53	a computation could be any kind of function X squared
13:57	could be the function three , X plus four could
13:58	be the function could be a linear function , could
14:00	be a quadratic function , could be a square root
14:03	as a function . All kinds of things can be
14:04	in the box . Now we're learning a new kind
14:06	of function basically , and it's called the inverse sine
14:09	or the inverse co sign . So basically what you
14:12	have is you feed numbers , not angles , just
14:15	numbers . You feed numbers into this box . Right
14:21	. And what is inside of this box ? This
14:24	box I'm going to represent right now is arc sine
14:27	or arc co sign or arc tangent . So it's
14:33	, I'm putting three different functions here as examples .
14:36	But the punch line is what comes out of the
14:39	box , angles come out of the box . It's
14:43	actually really easy to get confused . So I'm writing
14:46	it down here and of course you can get degrees
14:48	if you're working , injuries or radiance . So when
14:52	you're taking an arc sine or an art co sign
14:53	, you need to expect degrees coming out or you
14:56	need to expect pi over 63 pi over four radiance
14:58	coming out . Keep it in your mind because even
15:00	though I know that you know this it gets confusing
15:03	when you start doing art sign our coastline , all
15:05	this stuff that you expect angles to come out .
15:07	Sometimes we forget what we're doing . So numbers come
15:10	in . We're gonna talk a lot about the numbers
15:13	that are allowed to come into these boxes . The
15:15	boxes contain arc sine or cosine or tangent and the
15:18	output of these boxes are basically angles are getting degrees
15:21	or radiance . All right , so we're done with
15:23	the first part here . Now , the tricky part
15:26	is what's coming up next . The punchline , the
15:28	gotcha that I really want everybody to um to comprehend
15:32	. Let's go back to our example , sign of
15:34	some angle is one half . We already did it
15:37	. We said a sign of some angles . One
15:39	half means arc sine of one half is 30 degrees
15:42	or pi over six . Okay , so let's go
15:45	back to sign Of some angle is equal to 1/2
15:52	. Now remember back from basic algebra , you know
15:56	you have inverse functions , inverse operations to solve equations
16:01	. Right ? So if you have an equation like
16:03	three x equal six . You're multiplying on the left
16:07	hand side by three . So to get rid of
16:09	it , you do kind of the inverse operator .
16:10	And you divide opposite of multiplication is division . If
16:13	you have a square root you might undo the square
16:16	root by squaring both sides . So it un does
16:18	the square root , I can go on and on
16:20	. If you have a long algorithm on one side
16:22	then to undo the log rhythm , you might raise
16:24	both sides of the equation to a power of the
16:27	base of the law algorithm . To undo the logarithms
16:29	. So here's kind of a basic equation . I
16:31	know it looks so simple because we know that sign
16:33	of 30 is one half . So we can solve
16:35	this equation really easily . But really the real way
16:38	that you solve this equation is you undo the sine
16:41	function and try to solve for data and get the
16:44	function or the variable data by itself and then you
16:47	should have some kind of angle . As a result
16:50	we already know the answer . The answer is 30
16:51	degrees or pi over six . But really the way
16:54	that you sign it , the way that you solve
16:56	it is you have to undo the sign operation .
16:59	How do you undo and assign function ? You do
17:01	the inverse the arc sine to both sides of the
17:04	equation . So to solve this simple equation , what
17:06	we would do is apply the arc sine to both
17:10	sides of the equation , we apply it to the
17:12	left , which means we're applying the dark side to
17:15	the side guy . And on the right we have
17:17	to do the same thing to both sides . Just
17:19	like we have to do the same thing to both
17:20	sides of an equation all the time . So to
17:22	solve this equation , we already know the answer .
17:24	But basically we can apply the opposite functions of the
17:26	left and then apply the opposite function to the right
17:30	. All right now , what ends up happening is
17:32	the arc sine is the exact opposite of the sign
17:35	, so it under does the sign . And so
17:38	all you have left on the left is data and
17:41	on the right you have arc sine of one half
17:45	. And we already talked about the fact that our
17:46	sign of one half means that it's a 30 degree
17:49	angle . So then data 30 degrees or pi over
17:53	six , which we've already talked about . So I'm
17:56	introducing arc sine our coastline but I want you to
17:58	understand the big picture . We obviously want to go
18:01	backwards so that we can solve lots of times when
18:04	we want to figure out what the angle is .
18:06	But also because later we're gonna have trig equations which
18:09	are really big equations with trig and metric functions .
18:12	The goal of them is always going to be to
18:14	figure out what data is . So you have to
18:16	have a way to rip open that box and get
18:18	the data that's inside . And the way you do
18:20	it is to is to basically annihilate the sign with
18:23	its inverse , which is the dark side to annihilate
18:26	a co sign , you use an arc cosine function
18:29	to annihilate a tangent . You use an arc tangent
18:31	and we're applying it to both sides . Like we
18:33	solve all of these equation here . So we already
18:37	said the angle here is Pi over six . So
18:40	let's draw this and talk about it a little bit
18:43	and I'll use the unit circle that we have on
18:45	the other board as well . I think I want
18:46	to do it probably right over here . Let's draw
18:49	this function right here or this unit circle right here
18:52	, we'll do a little sketch . And I also
18:53	use the real unit circle that we have over there
18:55	. What were essentially saying is that if we go
18:58	to pi over six here here is about roughly speaking
19:01	a 30 degree angle , it's not perfect , but
19:03	this is roughly a 30 degree angle . So I'll
19:06	put feta this is pie Over six , which is
19:09	30°. . What we're saying is that the sign of
19:13	this angle of pi over six is one half and
19:16	that's why it is the solution of the equation .
19:18	Let's go check in on the inner circle . Here's
19:21	pi over six , which is 30 degrees . The
19:24	co sign is the first number of the sign is
19:25	the second number . So here we're saying the sign
19:28	of 30 is one half to sign up . Five
19:29	or six is one half , which means the projection
19:31	here is right because we're talking about the sign ,
19:35	the projections on the y axis , which is exactly
19:37	in the middle , which means it's at one half
19:39	exactly the case . But let me ask you this
19:42	, are there any other angles around the unit circle
19:45	that also have a sign that give you one half
19:49	? I mean think about what you're asking yourself ,
19:51	you're over here saying our sign of some angle is
19:55	one half . Give me give me the angles at
19:56	work and we figured out through doing all this arc
19:59	sine arc sine of one half , we're getting the
20:01	angles right and we got an angle five or six
20:03	we said , hey , it works , it's the
20:04	angle , that's the answer . But it turns out
20:07	there's tons of other answers , There's tons of other
20:09	angles that work because in order to solve this ,
20:12	all you need to do is figure out the angles
20:14	such that the sine of the angle is one half
20:16	here is one of those angles but walk with me
20:19	over here and look at this . If this is
20:21	pi over six , then right here would be to
20:24	five or six , then here would be 35 or
20:26	six , then here would be 45 or six ,
20:29	then here would be five pi over six . What
20:33	is the sign of this angle 55 or six ?
20:36	It projects onto the Y axis is exactly the same
20:38	location here , it's going to be positive one half
20:41	. So what we have figured out is that there's
20:44	actually multiple answers to this . Pi over six is
20:47	not the only answer , but let's just go through
20:49	it . It's a sign of pi over six is
20:53	indeed one half , so it does satisfy this equation
20:55	. However , the sign of five , pi over
21:00	six is actually also equal to one half . So
21:03	it looks like there's another answer that lives over here
21:06	. If you go back to the unit circle ,
21:07	the sign of this is one half . But look
21:10	at this , the sign of this angle is also
21:12	one half , because both of these angles project onto
21:14	the exact same location . But wait , there's more
21:19	, there's possible to have more angles . In fact
21:21	, there's an infinity number of angles that will give
21:24	you that same sign of one half . What happens
21:26	if I go all the way around the unit circle
21:29	and continue counting until I get here . What's gonna
21:32	happen there ? Let's just take a look . You
21:34	have five or 6 to 5 or 63 than uh
21:37	Four than 5 , 5 , 6 and six and
21:39	7 and eight . Then nine . Then 10 10
21:42	, 11 , 12 , 13 , 5 or six
21:44	is right here , 13 5 or six . So
21:47	we can say that sign of 13 pi over six
21:52	is also equal to one half . So this is
21:55	13 5 or six years , 14 , 15 ,
21:57	16 . Here's 17 pi over six . It gives
22:00	you the same projection here . So we also know
22:02	that sign of 17 pi over six is equal to
22:07	one half . So you see , I can play
22:08	this game again and again . Any time I land
22:10	here , any time I land here , I'm going
22:13	to get the sign of these angles . Both of
22:15	them are going to give me one half . Now
22:17	, if I land on angles down here , all
22:19	of these angles are gonna project to negative one half
22:23	. So the sign of those angles down there don't
22:25	work . Only the angles in the upper half plane
22:28	give me the sign the same exact sign with the
22:30	same exact sign . Once the positive one half .
22:33	Also notice you can go in the negative direction .
22:37	There's negative pi over six , negative 256 negative 356
22:40	negative 456 negative 556 negative 656 negative 75 or six
22:45	is right here negative 75 or six is the same
22:48	location . So the sign of negative seven pi over
22:52	six is also equal to one half . I encourage
22:54	you get a calculator or a computer and put all
22:57	these angles in there and then hit the sign button
23:00	. It's going to give you the same thing one
23:01	half and that's because it's a unit circle and I
23:04	can keep counting over and over again . Not only
23:07	can I keep going in circles to land on the
23:09	same place , but there's actually two different kind of
23:11	fundamental angles that also give me the same sign .
23:14	So the question is whenever I take the arc sine
23:17	of one half , which is what I'm doing to
23:19	solve this equation , what angle should I get back
23:22	? It seems like they're infinity number of angles .
23:25	They're an infinity number of angles . What angle should
23:27	the calculator return ? And this is where I come
23:30	back to what we talked about in the beginning that
23:34	certain , there's only a certain range of angles that
23:36	the calculator will return . So I need to walk
23:38	through . Why ? But in the back of your
23:40	mind , before we jump into all this uh stuff
23:42	that we have to talk about , just remember when
23:44	I take the ark sign of anything , I can
23:47	get an infinite number of angles because all I have
23:49	to do is figure out the fundamental ones . And
23:51	then I can keep spinning around the unit circle ,
23:53	finding an infinite number of additional angles for any arc
23:56	sine or any art co sign or any arc tangent
23:58	of any number , I can find tons of angles
24:01	that all work . So the calculator is not going
24:03	to give you an infinite number of answers . So
24:05	how does it know what to give you ? This
24:07	is what we have decided by convention in math .
24:10	The way it's gonna work , let's first talk about
24:12	only the inverse sine . It's called the inverse sine
24:15	. It's important for you to know that the sine
24:17	of an angle always lies between negative one and one
24:21	always . How do you know ? Because it's a
24:23	unit circle . The circle only has a radius of
24:27	one . So if I take the sign of any
24:29	number around here , it can only give me a
24:33	maximum up here . The sign up here would be
24:35	one positive one . Any angle over here will just
24:39	be a fraction of that projecting onto this axis ,
24:41	it'll be a fraction of that . The sign over
24:43	here is zero . The sign over here is also
24:46	zero . The sign over here is positive one and
24:49	the sign down here is negative one . Any angle
24:52	down here is going to be projected here . It
24:54	will be less than well , larger than negative one
24:57	, basically anywhere between plus one and minus one in
25:01	that range , larger than negative one and smaller than
25:04	positive one . It's impossible to get a sign go
25:07	type in any angle you want . Your calculator ,
25:09	hit the sign button , you will never get a
25:10	number larger than one or smaller than negative one .
25:14	Like you'll never get negative two or negative three or
25:16	negative four because it doesn't work like that the way
25:18	the projections work . Okay , So it's important for
25:22	, you know , that the input , that the
25:24	sine function can only spit out numbers between plus or
25:28	-1 . That means that the ark sign , which
25:32	is the opposite uh inverse . The opposite function of
25:35	the sign can only take as inputs to the bit
25:39	numbers between negative one positive one . Why ? Because
25:42	the arc sine is going backwards , I'm feeding numbers
25:45	in from the outside of the unit circle and I'm
25:47	getting angles back . So because the sign can only
25:51	give numbers between plus and minus one , then feeding
25:54	numbers into the arc sine . Remember this picture I
25:57	drew for you , feeding numbers into the arc sine
25:59	can only be between plus or minus one because those
26:02	are the only values that the sine function can give
26:05	out . Those are the only numbers on the outside
26:07	of the unit circle . So because of that ,
26:09	the only numbers that can go into this arc sine
26:12	are actually between plus and minus one . If you
26:14	go press let's say five , that's way outside that
26:17	range and hit arc sine . You'll get an error
26:20	. It won't do it because there is no angle
26:23	. It has a sign that gives you a five
26:26	or something because sign can only go between plus and
26:28	minus one . So the point is the inputs to
26:30	the arc sine can only be between plus or minus
26:32	one . But what I'm trying to show here is
26:34	the sign is as a projection on the y axis
26:37	. So the numbers between negative one on the y
26:40	axis down here and positive one on the y axis
26:43	here . Any number that I give into this function
26:46	is only going to be on this part of the
26:47	y axis between here and here . So the fundamental
26:51	angle , the basic angle that is returned from a
26:53	calculator is going to be the smallest angles that have
26:58	that as a sign . And so the smallest angles
27:00	would be from zero up to pi over to and
27:03	from zero down to negative pi over two . That
27:06	is by mathematical definition , those are the only angles
27:09	that are going to come back and that's going to
27:10	cover all possibilities . Well , there's still other angles
27:14	around the unit circle that will give you that sign
27:16	. But what I'm saying is the calculator is only
27:18	going to give you the fundamental angle back . We
27:21	all know that we can spin around the unit circle
27:23	and get additional angles that have the same sign .
27:25	But as far as what is that fundamental angle that
27:28	the calculator will give you back at the computer or
27:30	by definition of what the function is able to give
27:32	you back is always going to be in this range
27:35	. Because if you think about it , the sign
27:38	of , let's take some angles here . The sine
27:40	of pi over two is going to give you one
27:42	and the sign of negative pi over two is going
27:44	to give you negative one and the sign of zero
27:46	is going to give you zero . So any angle
27:49	that I pick in this shaded blue region is going
27:51	to give me a sign , appear at one or
27:53	a sign at negative one And the sign only goes
27:55	from -1 positive one . So by convention , when
27:59	I put negative one positive one in the calculator ,
28:01	it's not going to give me an infinity of angles
28:03	back , it's gonna give me the smallest angle possible
28:06	that satisfies the thing . So when we go back
28:08	over here and we said , hey , we're trying
28:10	to find what angle gives me . Some of the
28:13	sign is one half . It is true . There's
28:14	an infinite number of angles , but there's only one
28:17	fundamental angle and that's going to be an angle .
28:19	This first one it's going to be between negative pi
28:22	over two and pi over two . So when you
28:24	hit the inverse sine that's the angle you're going to
28:26	get back . You're never gonna get five pi over
28:28	six because five pi over six is over here .
28:31	123455 or six . It's outside the range over here
28:35	. You're only going to get that fundamental back angle
28:38	back . So that's the sine function . Let's go
28:40	have a similar discussion for the cosine function . The
28:43	coastline functions inverse . Cosine called our coastline . It
28:47	also can only give you values . The co sign
28:50	function can only give you values between negative one and
28:52	one . For the same exact reason any angle I
28:55	pick is going to project onto the X . Axis
28:57	. Which can only go between negative one and positive
29:00	one . So because of that the arc co sign
29:04	, the opposite function can only take his input values
29:07	between negative one and positive one inputs to the arc
29:10	cosine function but those negative one positive one . Those
29:13	are projections on the X . Axis before it was
29:16	projections on the Y axis . But for coziness projections
29:19	on the X axis . So those go from negative
29:21	one all the way to positive one along the X
29:23	axis . So these are the inputs to the function
29:27	. The output angles then are going to be the
29:29	smallest angles that give me that projection and the smallest
29:34	angles there is going to go from zero to pi
29:37	write these down here . These angles aren't gonna ever
29:39	give you a return value because um well you see
29:44	down here in any angle down here would project to
29:46	the same axis as this angle projecting down here .
29:49	So you have to have unique angles . I mean
29:51	think about it . If you sweep angles through here
29:54	, what's the the co sign of zero ? It's
29:56	going to be a one . What's a co sign
29:59	of uh pie ? It's gonna give you negative one
30:03	. Right ? What's the coastline of pi ? Over
30:04	20 So you see I've already said that the coastline
30:08	can only go between negative one and one . So
30:09	all of the angles required to do that up in
30:12	the top here because the co sign of any of
30:15	these angles here are going to go between negative one
30:17	and positive one . So the pot the punch line
30:19	is you can only feed negative one positive one in
30:22	. But the angles that come out are going to
30:24	be in the upper half playing up here between zero
30:26	and pi . It's important for you to realize when
30:28	you take an art co sign , hit the button
30:30	on your calculator . You will always get an angle
30:32	up here . You'll never ever get an angle down
30:34	here , even though those angles down there can still
30:38	give you the same co sign . We kind of
30:40	lock the function down into like a base angle return
30:44	. It's like the base angle that comes back ,
30:46	the most fundamental . The smallest angle that satisfies it
30:49	will be up here . So , so far arc
30:52	. Sine always gonna give you angles in the right
30:55	half plane like this between negative pi over two and
30:57	pi over to our co sign is always going to
30:59	give you angles back between zero and pi . Okay
31:02	, now the tangent function is the one that's hardest
31:05	to understand and I really want you very much to
31:08	understand it , so I'm going to have to do
31:10	a little bit more talking now . What's going on
31:12	here with the tangent ? Is that the tangent can
31:15	actually go between negative and positive infinity . Uh in
31:19	other words , the sign and the coastline . When
31:21	you take signing coastline of angles , you always get
31:23	between plus or -1 . However with tangent you can
31:26	get numbers way outside of that . Why ? Because
31:30	the tangent is the sign divided by the coastline .
31:33	It's the sign divided by the coastline . So let's
31:34	go through a couple of quick examples to remind you
31:37	of this . What is the tangent of negative pi
31:42	over two ? Negative pi over two . It is
31:45	the sign of negative pi over two Divided by the
31:50	co sign of Negative Pi over two . Right ,
31:54	so negative pi over two is down here , right
31:57	down here on the unit circle . So what is
31:59	the sign of that ? Well , the projection onto
32:01	the y axis is negative one . What is the
32:04	co sign down here ? Way down here , the
32:05	coastline is zero . Okay so what do you have
32:08	, you have negative 1/0 . And so what happens
32:11	is the tangent of this angle is actually negative infinity
32:14	. And if as you get away as you get
32:17	away from this negative pi over two like angles really
32:19	close , what happens is it gets really really ,
32:21	really big because the denominator is getting really really close
32:24	to zero . So go ahead and do it ,
32:26	go ahead and put an angle very close to negative
32:29	pi over two in but not quite there . And
32:31	you'll see the tangents like you know , 100 million
32:33	or something . It's because the top part is approaching
32:36	negative one and the bottom part is getting closer and
32:39	closer to zero . So we say at negative pi
32:41	over two , you actually get negative infinity . Uh
32:44	There for the tangent . Now let's take another example
32:47	. We took an angle down here . Let's take
32:49	the Angle zero . The tangent of zero radiance or
32:53	zero degrees . Whatever is the sign of zero divided
32:58	by the co sign of zero . What is the
33:00	sign of zero ? At zero here , there's no
33:02	projection on why ? So the sign of zero is
33:05	in fact zero . Okay . And what is the
33:07	co sign of zero ? The projections on the X
33:10	axis . The coastline is one . So because it's
33:12	0/1 , actually the tangent of zero is zero .
33:15	Certainly it's not infinity or anything . It's just a
33:17	number , it's it's zero . Right now let's go
33:20	back to the upper part . So we said this
33:23	is negative pi over two . This is zero ,
33:25	this is positive pi over two . Let's take a
33:27	look at what the tangent Of positive pi over two
33:31	is . It's the sign of Pi over two divided
33:36	by the co sign Of Pi over two . What
33:40	is the sine of pi over two will appear .
33:42	The sign is positive one positive one . What is
33:46	the co sign of pi Over two will appear to
33:48	co sign is zero . No projection onto the X
33:51	axis . So you get a zero positive over zero
33:54	, give you a positive infinity . So what I'm
33:56	trying to prove to you through an example is the
33:58	first time . It seems weird , students say ,
34:00	well , how can tangent go to positive and negative
34:02	infinity ? But Sine and Cosine can't . Sine and
34:05	Cosine are pure projections on the X and Y .
34:07	Axis of the unit circle , So they can never
34:10	be bigger or smaller than plus or -1 . But
34:12	tangent is not a projection . It well , it
34:15	is kind of but it's the division of two projections
34:17	. So because it's sign over coastlines , sometimes the
34:20	co sign on the bottom can go to zero .
34:22	And because of that it can drive the tangent sometimes
34:25	to negative infinity and sometimes all the way up to
34:28	positive infinity . So because of that , I've proven
34:30	to you the tangent function doesn't go between zero negative
34:34	one and positive one . It goes between negative infinity
34:36	, positive infinity . Start typing in angles into your
34:39	calculator . Hit tangent over and over two different angles
34:41	. You'll see that they go way outside the plus
34:44	and minus one range . So because of this ,
34:46	because tangent can actually give values between negative infinity and
34:50	positive affinity . What it means is the arc tangent
34:53	. The opposite operation can accept as an input value
34:57	values that are between negative infinity and positive infinity .
35:00	And those values are going into the function . And
35:03	then what I'm telling you here is that the angles
35:05	that come out of the function are also limited to
35:08	between negative pi over two and pi over two or
35:10	negative 90 and positive 90 . Why are those the
35:14	special angles ? Because remember the calculator or the computer
35:17	or whatever is going to give you the smallest range
35:19	of angles possible to satisfy the input . Kind of
35:23	like the input values possible . So if I put
35:25	negative infinity in or positive infinity in what are the
35:29	range of angles at work ? I actually just proved
35:31	it to you . If we put a negative pi
35:33	over two in for the for the tangent , we
35:35	actually get negative infinity . And if we put a
35:37	zero and we get something in the middle which is
35:39	zero . And if we put a positive pi over
35:41	two and we actually get a positive infinity . So
35:43	these angles here from negative pi over two up to
35:46	positive pi over to cover all the possibilities of where
35:50	tangent can go from negative infinity . Up to positive
35:52	infinity . Think about it . Anything from here .
35:54	On on up to hear the tangent here is negative
35:57	infinity . Make sure I'm right here . Yeah ,
35:59	the tangent here is negative infinity . The tangent here
36:02	is zero and the tangent here is positive infinity .
36:05	So these angles cover all the possibilities any number you
36:08	can think of sticking into a calculator , hitting inverse
36:11	tangent . These angles will cover it and they're the
36:14	smallest angles that will cover that range . So we
36:17	say the inverse tangent can take as an input any
36:20	number you want to put in and they can spit
36:22	out as an output anything between negative pi over two
36:24	and pi over two . Now , usually you see
36:26	up here I have the equal signs and I have
36:28	the equal signs . But for the tangent we don't
36:31	usually put the equal signs because we don't like to
36:33	deal with infinities in real life . So we say
36:35	the inverse tangent . I mean your calculator really never
36:38	gonna give you infinity back . I mean infinity is
36:40	an infinite number , it never stops . So we
36:42	really say the inverse tangent goes between negative pi over
36:45	two and pi over two as its output functions as
36:48	its output angles . Um But not exactly including like
36:52	it gets incredibly close to negative pi over two .
36:54	It drives it up super high basically to infinity all
36:57	the way up to positive pi over two . So
37:00	let me read my notes and make sure I didn't
37:01	forget anything . Anytime you take the inverse trig function
37:04	of a number , the calculator returns a base angle
37:07	which is like the smallest angle possible to satisfy the
37:12	smallest angle possible to satisfy what you gave it as
37:14	an input . Okay , other angles exist around the
37:17	unit circle that have the same sign or co sign
37:19	or whatever . But the calculator doesn't return all those
37:22	angles because there's an infinite number of them . It
37:24	returns the smallest angle possible , which are these ranges
37:27	I've written on the board here . So sometimes when
37:29	you're solving equations , what you'll do is you'll get
37:32	the base angle back The fundamental angle and sometimes you
37:35	might have to add or subtract 180° or 360° to
37:39	go into another quadrant depending on what the problem is
37:42	telling you to do and we will solve problems like
37:44	that later . But the fundamental base angle of what
37:46	your calculator will give you is going to be given
37:49	by these terms here . So what I want to
37:52	do , I think this was a really long lesson
37:54	, but I want to go through it all real
37:56	quick at lightning speed to make sure we're all on
37:58	the same page . Okay , so we know that
38:00	the sign of 30 degrees five or six is one
38:02	half . So we define an inverse going backwards where
38:05	the arc sine of the right hand side gives us
38:07	an angle back And so it gives us an angle
38:10	back . And we can also write it as the
38:12	inverse sine with the -1 . But that is not
38:14	an exponent . Don't treat it like an exponent because
38:16	it's not but you might see it in books like
38:18	this . The art co sign of a number gives
38:20	us an angle back , can be written in a
38:22	similar fashion . So numbers come in , angles come
38:24	out . Same thing with the arc tangent numbers come
38:26	in , angles come out and we can write it
38:28	with the -1 up there just like before . So
38:32	graphically numbers come in arc inverse sine Cosine tangent .
38:36	Arc sine cosine tangent are in here and angles come
38:39	out degrees or radiance . So then we start asking
38:42	ourselves the question , going back to our fundamental example
38:45	, sine of an angle is one half what angle
38:48	works ? So we apply the arc sine to both
38:50	sides , solve and we basically want to figure out
38:52	what the arc sine of one half is . That
38:53	means what angle exists So that I take the sign
38:57	of that angle and I get one half but you
38:58	know it's 30 degrees , you know it's pi over
39:00	six . So we put five or six there which
39:02	is right here in the unit circle . But then
39:04	we realized , wait a minute , there's other angles
39:06	because this angle also has the sign equal to positive
39:09	one half . And then we go even further and
39:12	say well wait a minute . Not only is the
39:15	sine of pi over six equal to one half and
39:17	the sign of five pi over six also equal to
39:19	one half . But if we spin around the unit
39:21	circle to here then we see the sign of 13
39:24	5 or six is one half . And if we
39:25	spin around the unit circle to hear another revolution ,
39:28	we see that 17 5 or six , the sign
39:30	of that's one half , even going negative to negative
39:34	seven pi over six , the sign of it ,
39:36	the projection is always in the same place one half
39:38	. So what angle does the calculator give you ?
39:41	That's what the rest of the lesson was about .
39:43	And we said that for the sine function it only
39:49	gives values back between negative one and positive ones .
39:51	So the inputs to the arcs and can only be
39:53	from negative one positive one which are projections on the
39:56	y axis between negative one and positive one . That
39:59	means the angles the smallest ones that will work for
40:02	this projection are always going to be between negative pi
40:04	over two and positive pi over two . Just like
40:06	this for the ark assign , it also gives numbers
40:09	between negative one and positive one . So the inputs
40:12	to the arc co sign is on the X axis
40:15	between negative one and positive one . These are the
40:17	projections on the X axis and the angles the smallest
40:19	set of angles that work are going to be between
40:21	zero and pi which is zero and 100 and 80
40:24	degrees . Then we said the tangents , the weird
40:26	one . It actually can take values as inputs between
40:29	negative infinity to positive infinity . Or I should say
40:31	the tangent gives values back between negative infinity and positive
40:35	infinity . So the input to the ark tan can
40:37	be numbers from negative infinity to positive infinity . And
40:40	we've already described why here it's because it's the ratio
40:43	of two trade function . So sometimes that denominator goes
40:46	to zero which drives the tangent to infinity . So
40:49	if the tangent can go between negative infinity positive affinity
40:53	, it turns out that the set of angles that
40:55	allow that to happen are also between negative pi over
40:58	two . Up to positive pi over two . Not
41:00	inclusive because we really don't want to get to infinity
41:03	. But anyway , those are the boundaries there .
41:05	Why is that ? Because the tangent of negative pi
41:08	over two is the sign over the coastline which is
41:10	negative infinity . And the tangent of positive pi over
41:13	two is the sign over the coastline which is positive
41:15	infinity . So this set of angles from here to
41:17	here is the smallest set of angles such that the
41:21	tangent goes from negative infinity to positive infinity . Now
41:24	that is a ton of talking , it's a lot
41:26	of talking and that's why every book I've ever opened
41:29	. It makes it incredibly hard to understand what they're
41:31	doing because it's hard to describe in words but fundamentally
41:35	that's what we're gonna do . So as we solve
41:36	problems with the inverse sine or the inverse co sign
41:39	or the inverse tangent , what we're gonna do is
41:41	try to figure out the angles that make the equation
41:44	work . We want to solve the angle because the
41:47	angle is what comes out . But the angles that
41:49	come out of an arc sine are only gonna be
41:51	in this range and the angles that come out of
41:53	anarcho sign are only going to be in this range
41:55	, and the angles that come out of an arc
41:56	tangent are only gonna be between this range . And
41:59	so we're gonna have to keep track of that as
42:01	we solve our problems and we will be doing that
42:03	in the future lessons . So make sure you understand
42:05	this as well as you can watch it a couple
42:07	times . If you need to follow me on to
42:08	the next lesson , we're gonna crank through a ton
42:10	of problems involving the inverse trig and metric functions .

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