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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