06 - Review of Essential Trigonometry (Sin, Cos, Tangent - Trig Identities & Functions) - Free Educational videos for Students in K-12

06 - Review of Essential Trigonometry (Sin, Cos, Tangent - Trig Identities & Functions) - By Math and Science

Transcript


00:0-1	Hello . Welcome back to the Physics one course .
00:02	This title of this lesson is called a review of
00:05	essential trigonometry . So every single physics problem that you're
00:10	gonna work is going to seem challenging to you in
00:12	the sense that no longer will the problem tell you
00:15	exactly what you need to do to find the answer
00:18	. So a lot of times , almost all of
00:20	the time when we get past the basic problems ,
00:22	there will be an angle of some kind in the
00:25	problem . Maybe it's a plane like a wedge where
00:28	a block is sliding down in the the plane has
00:30	an angle to it . Maybe a baseball is thrown
00:33	at an angle . Maybe you're pushing with a force
00:35	at 35° or whatever . I can go on and
00:38	on and on . There's almost always going to be
00:40	an angle in every problem , but you're never gonna
00:43	be told , hey , you need to do the
00:45	sign of the coastline of the tangent of this angle
00:47	. You're never gonna be told that you're going to
00:48	need to draw a picture and figure out from your
00:51	previous knowledge of trig in order what to do ,
00:54	you're gonna have to do that yourself and that's why
00:56	these problems seem difficult . So in this lesson we're
00:58	going to review basic essential trigonometry . It's important for
01:01	you to know that I expect that you have seen
01:04	this stuff before . If you haven't , then you
01:06	need to stop and go to my trade class ,
01:08	my trigonometry course and review the basic idea of angles
01:11	and stuff . And I am going to review as
01:14	we go through as we solve the problems , I'm
01:16	gonna baby step you as much as I can to
01:18	review you , but still you you need to have
01:20	been exposed to this material . So the first thing
01:23	we're gonna do the most important things these these ,
01:25	this is not a review of everything in trigonometry .
01:27	this is the review of the most important things to
01:30	solve physics problems . Right ? So everything in this
01:33	lesson is critical , fortunately none of it is really
01:36	hard . So if you have a triangle , remember
01:38	this guy , you talk about triangles all the time
01:40	in trigonometry , that's what it's all about . So
01:43	you have some kind of triangle and this is a
01:46	right angle which means it's 90° over in the corner
01:49	and there's some angle , like I told you ,
01:51	you might have a wedge a lot of times in
01:53	physics problems , we have a wedge and we call
01:54	this angle theta . So tha don't let it scare
01:57	you . It's just another variable like X . Or
01:59	why this could be 15 degree angle or this could
02:01	be a 35 degree angle or a 70 degree angle
02:04	or whatever , of course is the angle increases the
02:07	the steepness of this wedge gets taller or whatever .
02:09	But this is a general diagram here . Now ,
02:12	we label the different parts of this triangle . We
02:14	say that relative to this angle , the opposite side
02:17	of this guy , we're gonna label it opposite O
02:20	. P . P . For opposite in relative to
02:22	this angle . This side right here is adjacent .
02:25	Uh A D . J . We call the adjacent
02:28	. So adjacent means the side kind of close to
02:30	the angle opposite means the side opposite the angle and
02:33	then the very longest side of every right triangles called
02:36	the high partners . And I know that you all
02:37	know that from basic um from basic math . Now
02:43	, also in basic math , you learn something called
02:46	the Pythagorean theorem . Just , just don't forget Pythagorean
02:51	theorem . Don't forget that with right triangles like this
02:55	, you always have the pythagorean theorem at your disposal
02:58	. The physics problem will never say use the Pythagorean
03:01	theorem to solve it will never tell you that .
03:02	You just have to know that you can use this
03:04	for every right triangle . Pythagorean theorem . If the
03:08	way you learn geometry is C squared is equal to
03:10	a squared plus B squared . In this equation ,
03:14	C . Is always the hypotenuse , it's always the
03:17	longest side of this triangle , the other two .
03:19	Since they're added together , it really doesn't matter adjacent
03:22	or opposite . It doesn't matter what you label them
03:24	. So the way you read this , in terms
03:26	of this triangle that we have right here , since
03:29	C . Is the hypothesis will write it as hypotenuse
03:32	squared is equal to And again , I told you
03:35	it really didn't matter . So we'll just call it
03:36	adjacent squared plus opposite . Sure , what does this
03:43	equation mean ? That means for any right triangle ,
03:45	no matter what the angle here is Whether it's 15
03:48	degrees , 30 degrees seven degrees . Really slender triangle
03:52	, whatever . Doesn't matter if you know the length
03:54	of the hypotenuse when you square it and you know
03:56	the length of this and this and you square them
03:58	separately , and then you add these together , they
04:00	equal each other always . So it allows you to
04:02	find the third side of the triangle . Okay .
04:05	The reason I'm bringing this up is because we're gonna
04:07	use it A little bit later . But it's important
04:09	for you to know that applies to all triangles ,
04:11	not all triangles . All right triangles with a 90°
04:14	angle like this one here . Alright , The next
04:17	thing we're gonna do , we're gonna talk about the
04:19	fundamental trig functions . Now . In truth , there
04:26	are more trig functions than the ones we're going to
04:28	list here . But these are the trig functions that
04:30	are most used in physics . We come across anything
04:33	more complicated than this . I'll explain it as we
04:35	go , the first one I know that you all
04:37	heard of . It's the sign of some angle theta
04:41	. Now , in a real problem , you'll put
04:42	a number here like 45 degrees or something . But
04:44	we're drawing everything generally . So we say the sign
04:47	of whatever this angle here is is defined to be
04:51	the opposite side over divided by the high partners .
04:53	So you literally take however long the opposite side of
04:56	the triangle is and you divide by the high partners
05:00	, right ? That's defined as the sign of a
05:01	single . Notice that the sine of the angle involves
05:03	the opposite side . Remember that ? Because that's gonna
05:06	be important . The co sign of the angle of
05:10	this triangle is equal to the adjacent side . This
05:14	is divided by the hypotenuse again , notice that the
05:18	co sine of the angle involves the adjacent side of
05:21	the triangle . That's gonna be important . So remember
05:23	that . So just without going any further , the
05:25	sign of this angle involves something to do with the
05:28	opposite side of the triangle . The co sine of
05:30	the angle has something to do with the adjacent side
05:32	of the triangle . They both involve the hypotenuse ,
05:34	but that's kind of not so important for right now
05:38	The 10 . Well , it's important but it's not
05:40	gonna be um not trying , not trying to make
05:45	you remember that right now , the tangent of the
05:48	angle which actually we're not going to use quite as
05:50	much in physics problems . Most of the time you'll
05:51	probably use one of these , you'll see why as
05:53	we solve vector problems is defined as the opposite side
05:56	, divided by the adjacent side , so it's the
05:59	opposite , divided by the adjacent side . So these
06:05	are critically important if there's one thing that I would
06:08	write down in this review of essential trig . First
06:12	, it's called a review , I expect you have
06:14	seen this before and it's essential trick . Which means
06:16	it's not everything . It's the most important stuff for
06:17	these kinds of problems is I would want you to
06:20	know these three things . Sine cosine tangent , if
06:22	that's all you remember . That sign is opposite over
06:25	hypotenuse co signs adjacent over hypotenuse tangent is opposite over
06:28	adjacent . If that's all you remember , then you
06:31	can solve a great many problems because most everything else
06:33	that I'm going to talk about in this lesson is
06:35	just gonna be messing around with these functions to do
06:38	useful things . So , if you only have to
06:40	remember one thing , remember these and understand what it
06:43	means . I know a lot of you looking at
06:45	this . A sine theta , what does that really
06:47	mean ? It's literally just a fraction sign of whatever
06:50	this angle is . Don't worry about what the label
06:52	is . It just means that something that's related to
06:55	this angle is defined as a fraction defined on this
06:58	triangle . And if you forget which is which opposite
07:02	adjacent , whatever . Then just remember how to put
07:04	this picture together and draw this picture on your test
07:08	. For many years . I had to draw right
07:10	triangles and label everything so that I would keep it
07:12	straight now . I just remember it . But if
07:14	you need to draw that picture please do draw it
07:16	. Now we're gonna take this and we're gonna do
07:18	something practical with it on the other board here .
07:21	So let's solve a quick little problem . This is
07:23	not exhaustive , but we're just gonna do a quick
07:25	little problem just to illustrate what I'm talking about .
07:28	Here is a triangle like this . Now inside this
07:32	triangle , I'm going to again , I do not
07:35	know what this angle is . Sometimes I do know
07:37	what the angle is , but in this case I
07:38	don't . But what I do know now is that
07:40	this side is three m long , This guy is
07:43	four m long and this guy is five m long
07:47	and I'm saying this is a right triangle . First
07:50	of all , you can always verify this is a
07:52	right triangle . You should always make sure this is
07:54	a right triangle . How ? Because we know that
07:56	C squared is equal to a squared plus B squared
07:59	. This is true for any triangle that has a
08:01	90 degree angle in it . C . C .
08:04	Is the longest side . So that's five . Now
08:06	we have to square it and we're gonna check and
08:07	see if it's equal to this squared plus this ,
08:11	where notice that A . And B . Doesn't really
08:14	matter because see I did four squared plus three squared
08:17	. If I called a equal to three instead ,
08:19	then it would still be three squared plus four squared
08:21	. So the right hand side would be the same
08:23	no matter what . So the only one you really
08:25	care about is to make sure the longest side goes
08:26	into cease column here . So this is 25 this
08:30	is 16 and this is nine , So 25 equals
08:34	25 . Check if you ever get in inequality ,
08:38	like if you have 25 and this was 22 ,
08:40	then something's wrong because the triangle has to be a
08:44	right triangle in order to even use these sine cosine
08:47	and tangent definitions . And if this thing doesn't work
08:50	out , then this is not a right triangle ,
08:51	because the pythagorean theorem always is true for a right
08:54	triangle . Okay , so this was just an aside
08:59	. I wanted to show you the pythagorean theorem works
09:01	, but what I mostly want to do is I
09:03	want to ask you the question , what is the
09:08	sine of theta ? Mhm . So literally when this
09:12	happens , you go back to your definitions which we
09:14	just wrote now . So I'm gonna write it down
09:16	again . The sign of data is equal to you
09:18	. Remember , I want you to remember if there's
09:20	one thing in this lesson , remember that the sine
09:22	of an angle involves the opposite side . The co
09:25	sign of anyone involves the adjacent side . You're going
09:28	to find out why that's so important as we solve
09:31	problems . The sign involves the opposite side of this
09:34	triangle . And of course , it's divided by the
09:36	hot news . Okay , opposite side opposite to what
09:39	its opposite of this angle . So that means it's
09:42	three . The high partners is along the side ,
09:45	which is five . All right . So what this
09:49	means is that the sign of this angle is equal
09:52	to 3/5 of an exact fraction . If you put
09:55	that in your calculator , you'll find that 0.6 .
09:57	So if I asked you what is the sign of
09:59	fada you would circle this . The sign of data
10:02	is equal 2.6 . All right . Let's go work
10:05	the next part of this . Little simple problem .
10:07	And then I'll circle back and draw some conclusions at
10:10	the end . So that was for sine of the
10:12	angle . Let's calculate for part B . Let's calculate
10:18	the co sign . You know , let's do it
10:20	like this . Um Let's do like this . I
10:24	want you to find the co signed of the angle
10:27	. That's the question there . So what is the
10:29	co sign of the angle And I told you before
10:33	the co sign involves the adjacent side . So sign
10:36	involved , opposite co sign involves adjacent . Try to
10:39	burn that in your mind . So you're gonna write
10:40	that down adjacent and it's gonna be divided by the
10:43	same thing , the hypotenuse . So you go back
10:45	to your triangle . The adjacent sides of this angle
10:48	is four and your divided by the same high partners
10:51	. five . So it's an exact fraction 4/5 .
10:54	Yeah . Um so you would write down that the
10:56	co sine of the angle theta is exactly 4/5 and
11:00	if you want to put that on your calculator you
11:01	can 0.8 , that's the answer . So notice ,
11:05	first of all , before we go any farther ,
11:07	the sine of the angle came out to be some
11:09	kind of decimal less than one . And the co
11:12	sine of the angle also turned out to be some
11:15	decimal less than one . That's always going to be
11:18	true . Right ? So the sign when you ,
11:20	if you go take your calculator out right now ,
11:22	make sure it's in degrees degree mode because they're different
11:25	modes in the calculator , we're gonna be using degrees
11:27	in physics almost all the time . If you take
11:30	any degree angle you want , stick it in your
11:32	calculator , hit the sign button , you're always going
11:34	to get a number less than one . If you
11:36	stick any degree number you want any number you want
11:39	and put it in the calculator and hit the coastline
11:41	button , you're always going to get a number less
11:43	than one . That's gonna be important to understand when
11:46	I wrap this all up in a nice bow at
11:48	the end , but you should always get decimals for
11:50	these guys . Why ? Because they're fractions because we
11:53	know that the hypothesis is always the longest side of
11:55	this triangle and the hypothesis is always going to be
11:57	on the bottom , then whatever on the top is
11:59	always going to be smaller than the bottom . And
12:01	that means that the fraction will always be less than
12:03	one side and coast . I never get bigger than
12:05	one ever . That's the bottom line . Yeah .
12:08	All right . What is let's look at part .
12:10	See what is tangent of this angle ? Tangent is
12:14	defined as a mixture of the two . Tangent is
12:17	the one that involves the opposite side , but it
12:19	also involves the adjacent side . But there's no hypothesis
12:21	anywhere here . So we just write down the exact
12:24	thing that we we we know it's the opposite .
12:26	Alright . Almost type road down adjacent , it's the
12:29	opposite side . Over the hypothesis , opposite side is
12:34	three . Uh What's not over our partners , opposite
12:37	over adjacent . Sorry about that . Opposite over adjacent
12:42	, opposite is three adjacent is four . So 3/4
12:45	and you all know that that's 40.75 So you would
12:48	write down that the tangent of some angle . Theta
12:51	is 0.75 Or you could write it as 3/4 .
12:54	Let's find too . When their exact decimals like this
12:57	, I'm okay with decimals . So you can go
12:59	put it there . All right . So now we
13:01	know what the sine of the angle is , What
13:03	the coastline of the angle is , What the tangent
13:05	of the angle is also noticed . The tangent of
13:07	the angle is also less than one . Okay ,
13:09	for this example the tangents are less than one .
13:13	Now . It turns out that the tangent function is
13:16	not always in between . Uh is not always like
13:20	less than one or whatever . The tangent function can
13:22	go off the rails when you plot it . I'll
13:23	explain that some other time . But for sine and
13:26	cosine they always stay between plus one and minus one
13:29	. Okay . I didn't graphic for you but it's
13:31	plus one and minus one . This guy can kind
13:33	of go all over the place and um I don't
13:35	want to get into the reason why right now .
13:36	But that's just something you can begin observed when you
13:39	plot it . So let's find out the most important
13:43	thing here . What is the angle theta . In
13:47	other words , is this angle um 13 degrees .
13:50	Is this angle 17 degrees . Is this angle 35
13:53	degrees . What angle is it ? The first thing
13:55	I want you to know before we actually calculate the
13:57	angle is this angle is locked in place by the
14:01	distances of this triangle . In other words , if
14:03	you got some string out or a pencil and you
14:06	measured five and you measured four and then you went
14:09	up vertically and measured three . There would only be
14:12	one way that you could put those together to make
14:14	an actual triangle that connected that one triangle would have
14:18	an angle here . That would be fixed . It's
14:19	fixed because the size of the triangle are locked in
14:23	place and there you are arranging them in one way
14:25	because they have to make a right triangle . So
14:28	for any given set of sides , there's only one
14:30	angle . There's only one answer . I just don't
14:32	know . Is that 35 degrees ? Or is it
14:34	42 degrees ? What is it ? And so the
14:36	way I figure it out is there's lots of different
14:38	ways , but the first one is Let's go back
14:41	up here . We calculated what the sine of the
14:43	angle the sine of the angle here was , and
14:46	it was .6 . All right . So we know
14:49	that the sign of this angle is 0.6 . We
14:53	calculated that before . So remember from equations , from
14:58	equations like algebra equations you do the opposite to solve
15:02	for X right ? You might divide by something ,
15:04	multiplied by something . Ad by some to get X
15:06	by itself . But the angle that we want to
15:08	find is wrapped up in a sign . How do
15:11	we do that ? Well , we say , well
15:13	, the data is going to be equal to .
15:14	We have to do the inverse function . In other
15:18	words , we have to undo it with an opposite
15:20	. So if sign is the function inverse side ,
15:23	that's what the negative one means . This is inverse
15:25	sine or arc sine . You might see it un
15:27	does it . So like the opposite of addition and
15:30	subtraction . In the opposite of multiplication and division .
15:32	When solving equations , the opposite of squaring something is
15:35	a square root . The opposite of sine , inverse
15:38	sine , right . So when we do inverse sine
15:40	to both sides of this thing , in verse sign
15:42	on the left and does it . So we have
15:44	failed to inverse sine on the right . So what
15:46	we're writing here , So if you stick 0.6 in
15:48	your calculator and hit the inverse sine button , You're
15:51	gonna get the angle back and the angle in this
15:53	case is going to be 36.87 Degrees . So that's
16:00	the angle . So if you built this triangle ,
16:02	this would be 36.87°. . And it can't be any
16:05	other angle or else you can't make the triangle to
16:06	begin with . Now we use we calculated data using
16:10	the sign . Okay ? But we can actually calculate
16:13	it Using the co sign . Because we also learned
16:16	for this triangle that the co sine of the angle
16:19	is 08 . So if we want to get fatal
16:22	by itself , we have to inverse cosine both sides
16:24	. That eliminates that kind of annihilates the left hand
16:27	side leaving Fattah by itself . It will be the
16:29	inverse coastline . 0.8 . So if you put 0.8
16:32	in your calculator and find the inverse coastline button and
16:35	hit that one , what do you think you're gonna
16:37	get ? You're gonna get 36.87 degrees . 36 27°.
16:44	. Notice that matches exactly everything self consistent . Whether
16:48	you use the sign to find the angle or the
16:50	co sign to find the angle you get the same
16:51	thing . Now what do you think is going to
16:53	happen if we use the tangent ? So let's go
16:56	here . Just kind of squeeze it in the bottom
16:58	. If tangent of the angle is .75 , then
17:00	the angle should also be , what do you think
17:03	the inverse Tangent of 0.75 ? So if you put
17:08	0.75 and find the inverse tangent button , what do
17:10	you think you're gonna get ? You're gonna get 36
17:13	87 The purpose of this example was two things to
17:17	show you how to calculate sine cosine tangent and to
17:19	show you that you can take any one of these
17:21	things and do its inverse to find the angle .
17:24	Alright , Because you're gonna be finding the angle a
17:26	lot , a lot of times , it'll be here's
17:28	what the baseball is doing . What is the angle
17:30	? What angle did you throw it at ? Well
17:32	, eventually you're going to construct a triangle and you're
17:34	gonna end up having to figure out what this angle
17:36	is , and you'll use one of these . So
17:38	the second part of this is to tell you that
17:39	it doesn't matter which one you use , they all
17:41	give you the exact same answer . So there's not
17:44	so much a right way to find the angle .
17:46	It's just there's about 16 different ways . Not literally
17:48	, but there's a bunch of different ways to find
17:50	the angle . Okay . The only other caution I
17:53	want to throw at you when you're dealing with triangle
17:55	trig , like this is this angle is um When
18:01	you put these numbers in the calculator like .8 ,
18:03	find the inverse co sign , you do this inverse
18:05	tangent . So on when you do this stuff ,
18:08	the angle that your calculator is going to give you
18:10	is always going to be the positive angle . It's
18:12	always gonna be the angle as if the triangle was
18:13	drawn here and it's gonna give you this positive and
18:16	this is a positive angle because it's measured from the
18:18	X . Axis like this , this is X axis
18:21	Y axis . So this is a positive angle ,
18:23	right ? As we go and do more problems .
18:26	I'm going to caution you when you're inverse tangent thing
18:29	or inverse co signer inverse sine . You've got to
18:31	be a little careful because you have to know that
18:33	that calculators always gonna give you the positive angle because
18:36	sometimes what if I'm throwing the ball down , right
18:40	? So the triangle is not oriented up like this
18:42	, it's oriented down then that means that I'll actually
18:45	have a negative value for the why and a positive
18:48	value for the X . And so long story short
18:51	, you gotta be careful to look at the quadrants
18:54	of what you're actually doing when it returns that angle
18:57	, you just have to know that it's always going
18:58	to give you that positive angle . But your problem
19:01	might actually be throwing the ball the other way and
19:03	you might actually have to add 180° to that angle
19:06	or something like that to get the correct angle you
19:08	want . But when we get to that point ,
19:10	I will I will explain and caution you as we
19:13	do more problems like that for now , just know
19:15	when you inverse cosine inverse tangent , you're always gonna
19:17	get that positive angle back . That's all I want
19:19	you to remember at this point . All right .
19:23	So , um this is very important coming up next
19:27	if you have a triangle . Yes , as we
19:31	have been talking about and it has some angle theta
19:36	and it has an opposite side and it has an
19:38	adjacent side and it has a hypotenuse as they all
19:41	do . I'm kind of regurgitating over and over again
19:46	that we had talked about the fact that the sine
19:49	of the angle was equal to the opposite . Remember
19:51	sign deals with opposite over hypotenuse and we talked about
19:55	the fact that the co signing the angle had to
19:57	do with the adjacent side . Remember co sign deals
20:00	with adjacent over hypotenuse . Absolutely true . But there's
20:05	actually a little bit , I don't want to call
20:07	it easier , but it's a very , very useful
20:09	way to write this down . If you were to
20:13	take this and solve it for the opposite side ,
20:16	how would you do that ? Well , this is
20:17	a fraction . You have to multiply left and right
20:19	by the hypotenuse , right . So the if you
20:22	solve this equation , you're gonna see that the opposite
20:25	side is equal to the hypotenuse times signed data .
20:32	Mhm . Okay , let me do this one and
20:35	I'll explain what I'm talking about here . Also with
20:38	this one , the adjacent side is equal to the
20:42	hypotenuse times the co sign of data . It was
20:47	just these equations are the same thing as these .
20:51	There's no different . I'm not introducing anything new .
20:53	I'm just telling you that if you take this relation
20:55	and you solve it for the opposite side , you'll
20:57	have to multiply both sides by the hypotheses . So
20:59	it cancels on the right , giving the opposite side
21:02	here . You multiply by the high partners . Same
21:04	thing here here . You multiply by the high partners
21:06	. All right . Um These are very useful .
21:10	Okay . Uh They are the exact same thing .
21:16	Mhm as these . Of course they are but they
21:20	are very useful . And the reason they're useful is
21:22	because in physics we deal with something called a vector
21:25	. I'm actually gonna explain really briefly what a vector
21:27	is to you now , but then we're gonna have
21:29	an entire lesson actually , two or three lessons on
21:31	vector . So don't stress out if you don't get
21:33	everything right now . But the point is these particular
21:37	relations are very very useful . Um Let me explain
21:40	why they're useful because this is one of the very
21:42	first things you'll be doing in physics . Let's say
21:45	I throw a ball this direction , right ? How
21:50	fast do I throw it 10 m per second ?
21:53	Obviously I'm not throwing it horizontally , I'm throwing it
21:55	up with some angle . So what I do is
21:57	I define the angle and the way I define an
21:59	angle as I draw a right triangle , And I'm
22:01	always throwing the ball relative to the ground . This
22:04	is the ground here . So let's say I throw
22:07	that ball at 35°. . Let's put numbers in here
22:09	instead of just data . Okay , you all know
22:13	that if I throw a ball at an angle up
22:15	, I have to throw it kind of in two
22:17	directions . Right ? I'm when I do that I'm
22:19	throwing it horizontally , that's the horizontal speed . And
22:23	I'm also throwing it vertically straight up and down .
22:26	The mixture of those two motions horizontal and vertical is
22:30	what gives you the I know I know the path
22:32	curves , forget about the curving , but right ,
22:34	when it leaves your hand , It's going at an
22:36	angle which is a mixture of those two . And
22:39	that mixture is reflected in the triangle . You can
22:41	see that if I should throw it at 35°, ,
22:44	that the horizontal component of the velocity is bigger than
22:49	the vertical component of the velocity . So I might
22:52	have a vertical speed . Right ? I know I've
22:59	been throwing around the word speed and velocity . We'll
23:01	define these terms a little bit , but I think
23:04	all of you know , uh that what speed speed
23:07	, more or less is horizontal speed . Okay ,
23:12	So the reason I'm showing you and telling you that
23:14	these relationships are very useful is because of the following
23:17	thing . Usually I throw a baseball . This arrow
23:20	represents how fast I'm throwing that ball in total ,
23:23	in the angle direction . 10 m per second .
23:26	That's what that means . 10 m every second that
23:29	ball goes . That's pretty fast . That's faster than
23:31	I can throw . But anyway , it's a nice
23:34	number . Okay , But let's say that I want
23:37	to figure out what is the vertical part of the
23:38	speed . In other words , how much of the
23:40	speed exists only in the up and down direction ,
23:43	and how much of the speed exists only in the
23:45	horizontal direction . Because the angled speed is a mixture
23:48	of the two . All right , so here's what
23:52	you do . The vertical speed . notice in this
23:59	triangle the vertical speed is the opposite side of this
24:02	triangle . To the angle , notice that this equation
24:05	tells me the opposite side of this angle to opposite
24:08	side of the triangle is the hypothesis times the side
24:11	. Right . So I'm trying to find the opposite
24:14	side of this triangle , which I know from this
24:17	equation is the hypotenuse times the sine . Okay ,
24:23	Which means that the opposite side of this triangle is
24:27	equal to what ? 10 times the sine of 35
24:34	degrees . Right ? So the vertical speed Is equal
24:41	to if you take sign 35 and hit that on
24:43	your calculator and multiply the answer by 10 you get
24:46	574m/s . It's the same units as whatever I threw
24:51	it with . Let me kind of leave that alone
24:53	. Let's calculate the horizontal speed and then I will
24:56	wrap it all up together and and impress upon you
24:59	something that you really must remember . Okay so the
25:02	horizontal speed . Yes horizontal speed . What is that
25:08	? That's the according in this triangle , that's the
25:11	adjacent side of this triangle . That's what I'm trying
25:13	to find the adjacent side of this triangle . But
25:16	we just said from this equation from the definition of
25:18	co sign , I can find the adjacent sign of
25:20	any triangle hypotheses times co sign . So it's the
25:23	hypotenuse Times The co sign of . Now I know
25:27	the angles 35° right . And so let me switch
25:32	colors too . Kind of put the point home so
25:35	the horizontal speed is equal to what it's one of
25:42	the hypothesis here is 10 . So let me go
25:44	and just write it down here . So it's 10
25:46	times the co sign of 35 degrees . So if
25:49	you take 35 degrees , hit the coastline but multiplied
25:51	by 10 , you'll get 8.19 meters per second .
25:57	Yes . Now , first of all , let's see
25:59	if this makes sense . We're saying that we're throwing
26:02	a baseball at 10 m per second at some angle
26:04	. And since the angle is pretty small , less
26:07	than 45 degrees , were saying just from the triangle
26:09	, the way it's drawn , the horizontal speed should
26:12	be larger than the vertical part of the speed .
26:14	The horizontal speed should be larger than the vertical part
26:17	of the speed . Uh And that's true . So
26:21	why did I bring this up ? Because the probably
26:23	the most important thing in trigonometry that you will learn
26:26	in physics and this is why I'm turning around and
26:29	I'm looking right at you because I really really want
26:31	you to remember this is we use triangle trigonometry to
26:35	take what we call vectors and break them up into
26:38	what we call components . This 10 m/s is called
26:42	a vector . Please don't let the words scare you
26:44	, vector is not a complicated thing . It means
26:46	I throw a ball at a certain speed . I
26:48	mean you know that that's not that hard to know
26:50	. The length of this arrow is 10 . Okay
26:53	, but maybe I don't want to just know what
26:55	that angled length is . I want to know how
26:57	much is in this direction , horizontal and how much
26:59	is in this direction , vertical . So we take
27:02	that vector and we break it into components meaning a
27:04	horizontal direction in a vertical direction . And we use
27:08	sine cosine tangent every single time to do that .
27:10	Mostly sign and co sign to do that . Okay
27:13	, So here's the equations that way I taught you
27:15	and then we apply down here . The opposite side
27:18	of a triangle is hypotenuse time sign . The adjacent
27:21	side of the triangle here is hypotenuse times coastline ,
27:24	hypotenuse is the total speed in this example , 10m/s
27:28	. And I'm throwing the baseball . Here's what I
27:30	have written in my notes . I'm not gonna write
27:31	it down , I'm just gonna say it about four
27:33	times to make sure you understand when we use a
27:37	sine function and multiply it by the total length of
27:41	the baseball . Uh speed . In this case ,
27:44	what the sign is doing , listen to me here
27:47	is it's taking the total speed 10 m per second
27:51	and it's chopping it is what I have actually .
27:53	This is the way I think about it . It's
27:55	chopping it down . You throw the ball at 10
27:57	m per second , but I want to chop it
27:59	and I want to know only how much is going
28:01	in the vertical direction , that's the opposite side going
28:03	up . So when you take a sign of an
28:06	angle and multiply it by the hypotenuse what you're doing
28:09	is you're taking the total speed and you're chopping it
28:12	finding how much of it goes in the vertical direction
28:15	because vertical is opposite . Remember I told you Sign
28:18	always deals with opposites and co signs always deal with
28:21	adjacent . And that's to help you remember that .
28:24	When you take the sign of this angle , multiply
28:26	it by this , it takes this number and it
28:28	chops it only to give you the vertical part of
28:31	the speed , right ? And when you take this
28:33	number and take its co sign that deals with with
28:35	adjacent . So when you take the coastline of this
28:38	and multiplied by the same thing , you're chopping it
28:41	also , but only giving the horizontal part , which
28:43	is this . So here's the moral of the story
28:45	, because in this case I've given you a speed
28:48	at 10m/s at an angle . But this vector thing
28:51	we haven't talked much about , it can represent lots
28:53	of things . It can represent a force . Maybe
28:56	I'm pushing on something at an angle and I'm pushing
28:59	with £100 of force or 100 newtons of force .
29:01	We'll talk about Newton's later . It's the unit of
29:03	force . Okay . But I want to know not
29:06	. How much am I pushing at an angle ?
29:07	How much am I pushing horizontally ? And how much
29:10	am I pushing vertically ? Well , I do that
29:12	by using triangle trig . I take this when I
29:16	multiply it by the co sign of this . It
29:18	chops it and gives me only the horizontal part when
29:21	I take this and I multiplied by the sine of
29:23	this . It chops it and it only gives me
29:25	the vertical . It's crucially important because all these physics
29:28	problems , what we're gonna do , we're gonna break
29:30	them all apart into X . Direction . And we're
29:33	gonna solve those and then we're gonna break them separately
29:35	into Y . Direction and we're gonna solve those .
29:37	We're gonna break everything apart and solve the different directions
29:40	separately . It's important to be able to take this
29:42	and chop it this way and chop it this way
29:44	to get the two different components . Mhm . So
29:47	here is your mega mega mega summary of everything we
29:50	learned in this section , summary summary . All right
29:57	. Here's the summary when we have a triangle like
30:00	that . The sine of an angle is the opposite
30:04	. What's the opposite ? Over the hypotenuse ? The
30:09	cosine of an angle is the adjacent . Over hypotenuse
30:14	the tangent of an angle is the opposite over the
30:20	adjacent . So it doesn't involve a hypotenuse there .
30:23	Okay , now we can take these guys and we
30:26	can we can find use these equations to find the
30:30	angles and every one of these cases . How do
30:31	we do it ? We can say that the angle
30:34	is equal to the inverse sine of the exact same
30:38	thing opposite over hypothesis . We can say that the
30:43	angles also equal to the inverse cosine of the adjacent
30:48	over the high partners . And we did an example
30:50	showing exactly that . And we can say that this
30:53	angle is also equal to the inverse tangent of opposite
30:58	over adjacent . Yeah , literally you take the fraction
31:02	, stick it in there if you want to find
31:03	the angle , you just inverse it . That's all
31:04	you do . Okay , which one of these is
31:07	correct . They're all correct . You can find the
31:09	angle using any one of these relations and we actually
31:11	showed you that a minute ago . And then the
31:14	final thing I want to point out is that we
31:16	can take these equations . So that's a useful set
31:19	of relations . We can take these set of equations
31:21	and we can write even more useful set of relations
31:24	. We can say that the opposite side of the
31:26	triangle is just the hypotenuse times the sine of the
31:31	angle that's going to give you the vertical . If
31:35	you kind of think of your triangle , the way
31:37	we've been drawing it and the adjacent side of that
31:40	triangle is the hypotenuse times the co sign of the
31:44	language . The sign chops this thing to give you
31:47	the vertical part and the coastline chops this thing because
31:49	you're multiplying by the high partners giving you the horizontal
31:53	part . So when you look at a trig book
31:57	or physics book , you'll see these equations , you'll
31:59	see these equations and you'll see these equations . And
32:01	it gets very confusing . Which ones do I remember
32:03	? Really all you really need to remember are these
32:06	these are the core equations . These just come from
32:10	these , these just come from these . So really
32:12	just remember these . But what's gonna happen is you're
32:14	gonna use these so much that you will remember them
32:18	to find the vertical part of a vector . You're
32:20	gonna take the hypothesis and chop it with a sign
32:23	, chopping with a sign gives you that vertical part
32:25	to find the horizontal part of some vector . Some
32:27	arrows hypothesis of a triangle are gonna take the hypothesis
32:30	and you're gonna chop it with the co sign that
32:32	gives you the adjacent to the horizontal part . Those
32:35	are the core things of trig . When you take
32:38	a trade course , you'll do so much more than
32:39	what we've done here . You will plot sine cosine
32:42	and tangent as you'll see that sign and co sign
32:44	go up and down like waves and you'll see that
32:47	tangent looks even crazier and there are other trig functions
32:49	besides sine cosine and tangent that you learn in a
32:52	trade course in a trade course . You'll also learn
32:54	or a pre calculus course , You'll learn about the
32:56	unit circle and you'll learn about how to calculate sine
32:58	cosine and tangent using the unit circle and you'll learn
33:00	about radiance and you'll learn how to convert radiant measure
33:03	two degrees because that's a unit of measure of ,
33:05	of angle also . But in physics which is what
33:08	I'm focusing on . These are the most important things
33:11	. These things are important . If you don't understand
33:14	this , watch it a few times . If you
33:15	don't understand anything about it then go review , trade
33:18	using my trig and calculus class and then come back
33:21	here and continue learning physics with me . We'll do
33:23	everything step by step and give you plenty of practice
33:25	to do really well and get a deep understanding of
33:28	the topics .

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