07 - Trig Functions of Acute Angles - (Sin, Cos, Tan, Cot, Sec & Csc Theta) - Part 1 - Trig Ratios - Free Educational videos for Students in K-12

07 - Trig Functions of Acute Angles - (Sin, Cos, Tan, Cot, Sec & Csc Theta) - Part 1 - Trig Ratios - By Math and Science

Transcript


00:00	Hello . Welcome back . The title of this lesson
00:02	is called trig and metric functions of acute angles .
00:05	This is part one . Really complicated sounding title but
00:08	I promise I'll break it down for you so it
00:10	will be very easy to understand . So we have
00:12	to introduce all of the trig and metric functions all
00:15	together . In the last couple of lessons , we
00:17	have already introduced sign and we've already introduced co sign
00:20	. And we've done a ton of discussion about what
00:21	those actually mean . So now we have to really
00:24	take Sine and Cosine and extend those ideas to the
00:28	other four trig functions because really there are six trig
00:31	functions in all . But the good news is you
00:33	don't have to memorize too many things because really the
00:35	fundamental trig functions are the Sine and the Cosine functions
00:39	. Everything else . All of these other trig functions
00:42	come from sine and Cosine . So verbally I'll just
00:45	go through it with you here before we write anything
00:48	down . We have sign , we have co sign
00:50	. We also have tangent then we have to define
00:53	something called co tangent then seek it and then Co
00:57	secret . So Sine Cosine tangent . Co tangent seek
01:00	it and Co secret . So that's six altogether .
01:03	But as I said the fundamental ones really are just
01:05	sign and co sign . All of the other ones
01:07	come from signing co sign . So I'm gonna write
01:10	a ton of stuff down and it's going to look
01:12	intimidating . But in the back of your mind I
01:14	really need you to understand that really . All it's
01:16	important or the most important is Sine and Cosine .
01:19	That's why I spend entire lessons on them just a
01:22	minute ago . Everything else comes from that . Okay
01:25	so these are trigonometry functions of acute angles . That
01:27	just means acute angles are angles less than 90 degrees
01:31	. And so that's how we're gonna start the discussion
01:32	. So at first we're going to start learning about
01:35	these trig functions of acute angles which are just angles
01:38	smaller than 90 degrees . But just realize in the
01:41	back of your mind that as we go through the
01:42	lessons will be extending it to what happens with the
01:45	angle is larger than 90 degrees . All these crazy
01:48	large angles we already introduced , we're going to know
01:50	and have to learn how to take trig functions ,
01:53	signs and co sines and tangents and so on of
01:55	all of those angles too . But we have to
01:57	crawl before we walk . So we're going to talk
01:58	about acute angles . All right . So what I'm
02:00	gonna do is draw a triangle on the board and
02:02	I'm gonna write all the trig functions down and then
02:04	we're going to talk about them and you should understand
02:06	exactly where they come from by the end of this
02:09	. So everything boils down to what a triangle is
02:12	. So we're gonna draw a right triangle . When
02:14	I say triangles . Everything boils down to triangles ,
02:17	I mean right triangles . Okay , so what I'm
02:19	going to do is draw a little triangle there and
02:22	then we need to kind of superimpose the triangle on
02:24	an X . Y axis . So the Y axis
02:28	is going to that's a terrible y axis . Let
02:30	me see if I can clean that up a little
02:31	bit . The y axis is going to be going
02:35	straight up and down a little better , not too
02:36	much . Uh and then the X axis is going
02:39	to be going across like that . So this is
02:41	the X axis , you all know , this is
02:42	always the Y axis and there is some angle .
02:45	This uh this ray right here uh is some angle
02:51	theta to the X axis . Remember all angles are
02:54	measured with respect to the positive X axis . So
02:56	this is just the angle measure right there . Now
02:59	the tip of this ray right here , they're formed
03:01	some point right at the end here . And so
03:04	this point has some X and Y coordinate over here
03:08	. I don't know what they are . It doesn't
03:09	matter . But every point in the xy plane has
03:11	an X . Y coordinate . So we're just saying
03:13	there's a point right there at the end . We
03:15	call it P . It has some coordinates X and
03:17	Y . And when you connect this point to the
03:19	origin , they formed some angle theta right here .
03:21	All right now we have to talk about this triangle
03:24	which is the black thing . Now , remember ,
03:25	this is a right triangle because we have a right
03:27	angle right here . All right . So what we
03:30	need to do is label a few things . Now
03:33	the hypotenuse of the triangle is always opposite from the
03:36	right angle . So , we're gonna call this H
03:38	Y . P . That means high pot news ,
03:41	right ? But also we're going to label it uh
03:44	in a different way as well . So , I'm
03:45	gonna do a little curly brace like this and tell
03:49	you that this length right here , we're gonna call
03:51	it our you in different books , might see different
03:55	letters . It doesn't matter what your book has or
03:57	exactly the notation all I'm trying to tell you is
04:00	every triangle that's right has a hypotenuse . And the
04:02	hypothesis sometimes you call it C . For pythagorean there
04:05	and we're just gonna call it are here . So
04:08	that's what we're doing . And uh so we have
04:10	a hypotenuse here . Now this angle has a side
04:13	of the triangle that is opposite to it . That's
04:15	this side and an adjacent side of the triangle here
04:17	. So this side is what we call the opposite
04:20	side . O . P . P means opposite ,
04:22	and this side of the triangle is called the adjacent
04:25	side . So when you look at an angle ,
04:27	you need to kind of figure out where the hypothesis
04:29	is , and when you go down to the side
04:32	closest nearest this angle here , that's called the adjacent
04:35	side , and the side opposite to that angle is
04:37	called the opposite side . Okay , so the opposite
04:41	side is basically if you think about it , the
04:45	opposite side of this triangle goes from this point P
04:47	down to the axis right here . So really if
04:50	I kind of get my curly brace in here ,
04:52	like this , something like this , this distance is
04:56	why units above the axis like this and then this
05:00	distance right here , this distance is X . The
05:03	reason I'm drawing all this stuff is because usually when
05:06	you first learn trig functions , you talk about opposite
05:09	and adjacent sides and that's handy for a bit .
05:12	But you eventually dropped that and you don't talk about
05:14	opposite and adjacent . When we get a little farther
05:17	into the class here we talk about the X .
05:19	Side and the Y side and all of this .
05:21	So I'm drawing everything on one figure , I'm saying
05:24	, hey , every triangle has a hippopotamus . We
05:25	also call it our every triangle has an opposite side
05:29	. We also call it why the reason we call
05:31	it wise , because there's why distance units from here
05:33	to the tip of this triangle . Okay , every
05:36	triangle also has an adjacent side adjacent to this angle
05:38	. We also call this side X . Because it
05:41	is X distance units from here to here . So
05:43	X , Y and R . So that's what we're
05:45	basically doing . All right now , what we're gonna
05:47	do is use this channel to define the six trillion
05:50	metric functions . And then I'm gonna show you a
05:52	quick way to understand how to remember them all .
05:54	So I really only need you to understand sign and
05:57	co sign . Uh Honestly , the rest will just
05:59	fall out of the discussion and then we'll solve of
06:01	course some problems toward the end and a bunch of
06:03	problems as we go through the lessons here to make
06:05	sure you're comfortable with it . All right . So
06:07	the first trig function that we must must must understand
06:11	. We've already introduced it in the last lesson .
06:13	It is called sign of this angle theta . So
06:17	sign of this angle Theta is defined to be the
06:20	opposite side of this triangle , divided by the hypotenuse
06:23	of the triangle . So it's the opposite side over
06:26	hypotenuse in every textbook that you open , whether it's
06:30	geometry or algebra or calculus or pre calculus or trick
06:33	or whatever , you'll always see a triangle and it
06:35	will have an opposite side and adjacent side . And
06:37	you'll see that the sine function is defined to be
06:39	the opposite side of the triangle , divided by the
06:42	hypothesis . That means if I actually measure if I
06:45	took a ruler and measure this length and then measure
06:47	this length and divided them . That would be what
06:49	we call sign of this angle now , because the
06:52	opposite sign is also called why we can put a
06:56	Y . Up here . And the hypotenuse is also
06:58	called are we can put an R . Down here
07:00	? So the sine of the angle is why over
07:02	our it's just the opposite over the adjacent side .
07:05	Right now , the co sign we also introduced in
07:08	the last lesson and that was defined to be the
07:11	adjacent side of the triangle divided by the high partners
07:14	. So if you take this side of the triangle
07:15	, measure it and however many centimeters it is .
07:17	And divide by this , then what you'll get is
07:20	what we call the co sign . Now , since
07:22	the adjacent is the X . Variable X . Label
07:26	. We can say this is X . Over our
07:28	. So you see we use this opposite adjacent hypotenuse
07:31	business at first , but eventually we're gonna drop all
07:33	that . We're just gonna start talking about X's and
07:35	wise and ours . So in your mind you need
07:37	to have this triangle burned in . So you know
07:40	what that actually is . And that's why I'm drawing
07:42	it here . Now in the last lesson , we
07:43	spent a tremendous amount of time talking about what sign
07:46	and co sign . Really mean if you haven't watched
07:48	that lesson , I actually really , really hope you
07:51	pause this . Go watch it first . But I'm
07:53	gonna summarize what we did in about 30 minutes in
07:56	the previous lesson , took 30 minutes to go over
07:57	it in great detail and I'm gonna summarize it in
07:59	the following way . When you see sine of an
08:02	angle , what you really are talking about is it
08:05	is the ratio , it is the ratio of how
08:07	much this triangle goes up compared to that means division
08:11	compared to how much total length of the hypotenuse of
08:15	the triangle there is and I told you in the
08:17	last lesson that we called this the chop factor four
08:26	, why the reason we call it the chop factor
08:30	and that's my word . That's not a word that
08:31	you're going to see in a book anywhere is because
08:35	it is literally a decimal that comes out , it's
08:37	a number less than one . When you take Y
08:39	and divided by . Are you get a number less
08:41	than one ? And it tells you in a number
08:44	for me how much of this triangle is going up
08:47	in the Y direction compared to the total length of
08:50	the triangle , which is the hypotenuse . So if
08:52	the chop factor for why is very big then that
08:56	means that this , why is this triangle is really
08:58	tall and most of the triangle is going up okay
09:01	, but if you have , I'm going to get
09:03	to in a second but the chop factor in the
09:05	X direction , if you have a large chop factor
09:08	in the X direction , it means that most of
09:10	the triangle is actually going along the X direction .
09:12	So these chop factors my words , but sign basically
09:16	tells you how much of the triangle is going in
09:18	the vertical direction , in the Y direction . Is
09:20	it a really tall triangle in the Y . Direction
09:22	? And if you have a very small chop factor
09:25	in the Y . Direction , it means that that's
09:26	not the case and the similar thing for here .
09:28	So we said in the last lesson that this co
09:31	sign is the chop factor for the X . Direction
09:39	, right ? Because we learned in the last lesson
09:42	that if we know what the sign and the co
09:43	sign is of an angle and there's a button on
09:46	your calculator , we're gonna learn how to use it
09:47	later . But if you know what the sine and
09:49	cosine of an angle is , then you know what
09:52	these chop factors are . Sign goes with why it
09:55	is the chop factor in the Y . Direction and
09:57	co sign goes with X . That's the chop factor
09:59	in the X . Direction . And so if you
10:01	know what these chop factors are , then I can
10:03	take the chop factor and multiply by the high pot
10:05	news . And that will tell me how many units
10:08	my triangle is in the Y . Direction . If
10:10	I'm multiplying by this one and if I multiply by
10:13	this one , it's telling me how many units it
10:16	is in the X . Direction . In other words
10:17	, if I know that my triangle has I partners
10:19	of 10 m . If I multiply by the the
10:23	sign which is the chop in the Y . Direction
10:25	that I'm gonna get this side of the triangle because
10:27	look it's opposite over hypotenuse if I then multiply by
10:30	the hypotenuse , the hypotenuse will cancel and I'll be
10:32	left with the opposite side . That's why it's called
10:35	the chop . Because you when you multiply by the
10:38	hypothesis , if you multiply the sine of the angle
10:40	. Times a hypothesis it cuts the hypotenuse down and
10:43	only tells you how much of it goes in the
10:45	vertical direction . If you take the chop in the
10:48	X . Direction and multiplied by the high pot news
10:50	, then what your understanding is the answer is it's
10:52	chopping down that hypothesis and telling you how much of
10:55	this hypothesis exists in the X . Direction . So
10:58	I call it chopping X . And chopping . Why
11:00	? Because when I multiply those factors times the high
11:03	pot news . If I multiply the hypotenuse times the
11:05	sine , it gives me the length of the opposite
11:07	side . If I multiply the length of the hypotenuse
11:10	times the co sign of this angle . It gives
11:12	me the length of the adjacent side and that we're
11:15	gonna learn through solving many many problems . That's why
11:17	I call it the chop factor . Another way you
11:19	can look at it is it's a projection . If
11:22	I shine a light in this direction , the shadow
11:24	that the hypotenuse creates over here against a screen .
11:28	If I put a screen here would be the length
11:30	of this triangle . So the sine of the angle
11:32	is basically telling you , since it's a ratio of
11:34	why to high pot news , it's telling you how
11:37	much of that shadow is going to exist over here
11:41	. And so it's kind of a projection of that
11:43	hypotenuse onto a screen over there in the Y direction
11:46	. Or if I put a light vertically underneath like
11:49	this , it would cast a shadow down there .
11:50	So the sign is kind of a projection in the
11:53	Y direction of the hypotenuse . And the co sign
11:55	is the projection of the hypothesis in the X direction
11:59	. So whether you think chopping X and chopping Y
12:01	. A projection and ex projection and why ? It's
12:03	all the same thing . The sign goes with the
12:05	Y direction . The co sign always goes with the
12:09	X . Direction . And that is something you must
12:11	remember . All right , So now that we know
12:13	what Sine and Cosine are and we've kind of revisited
12:16	what we already learned in the past . We need
12:17	to talk about the tangent function . So the tangent
12:20	of some angle theta is defined as follows the opposite
12:25	side divided by the adjacent side , so the opposite
12:28	side divided by the adjacent side . Like this .
12:33	So the opposite side is why ? And the adjacent
12:36	side is X . So if I put it in
12:37	that notation , it will be Y over X .
12:40	Now we said that the sign is the ratio of

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