14 - Reference Angles Explained - Sine, Cosine & Unit Circle - Part 1 - By Math and Science
Transcript
| 00:00 | Hello . Welcome back . The title of this lesson | |
| 00:02 | is called reference angles Part One . Now Truthfully this | |
| 00:06 | is one of the most important topics of the sequence | |
| 00:08 | because it's going to allow us to calculate the sign | |
| 00:11 | in the coastline of angles anywhere around the unit circle | |
| 00:14 | . So the title reference angle doesn't sound important but | |
| 00:17 | actually it's really important if you remember we've talked about | |
| 00:20 | quadrant one quite a bit . We've talked about how | |
| 00:22 | to find the sign and co sign Of angles and | |
| 00:24 | quadrant one . We drew a chart and all of | |
| 00:26 | that . And I just said hey when we get | |
| 00:28 | to the other part of the unit circle , all | |
| 00:29 | the way around , we're gonna get to that later | |
| 00:31 | . Here is where we start to get to that | |
| 00:33 | . So it's important that you understand this concept in | |
| 00:36 | order to be able to take sine cosine tangent all | |
| 00:39 | the way around of any angle . So what I | |
| 00:41 | want to do is read the definition of a reference | |
| 00:43 | angle . I don't want to write it down because | |
| 00:44 | it's very simple to understand and we'll just take too | |
| 00:46 | long to write it but we'll talk about it and | |
| 00:48 | we'll do a ton of problems . Okay , so | |
| 00:51 | the reference angle is called theta prime . So usually | |
| 00:54 | we use the greek letter data to represent an angle | |
| 00:58 | . Theta prime is a reference angle . It's related | |
| 01:01 | to data but it's a different angle is called Data | |
| 01:03 | prime . It's the smallest acute angle between the terminal | |
| 01:07 | side of an angle and the X axis and is | |
| 01:09 | denoted data prime . Data prime . It's always positive | |
| 01:13 | and it's always a cute now if you're like me | |
| 01:15 | , you read this the first time , it makes | |
| 01:16 | no sense . So what I'll do is I'll show | |
| 01:19 | you on diagrams exactly what it is . Very simple | |
| 01:21 | . Read it one more time . A reference angle | |
| 01:23 | . Data Prime is the smallest acute angle between the | |
| 01:27 | terminal side of an angle and the X axis . | |
| 01:30 | So basically what you want to do is look at | |
| 01:32 | the angle no matter where it is , and figure | |
| 01:34 | out how many degrees exist between that angle and the | |
| 01:37 | X axis anywhere around the unit circle . You're always | |
| 01:40 | going to get a positive number for the reference angle | |
| 01:43 | . All right . So let's talk about it before | |
| 01:44 | we solve some problems . Just talk through it a | |
| 01:47 | little bit reference angle , smallest angle between the terminal | |
| 01:50 | side of an angle on the X axis . So | |
| 01:52 | let's say you had a 30 degree angle . This | |
| 01:54 | was data and I ask you what is there to | |
| 01:56 | prime the reference angle . Well in this case this | |
| 01:59 | angle is 30 degrees up as measured from the X | |
| 02:02 | axis . So the definition says it's the smallest acute | |
| 02:04 | angle between the terminal side . Remember this is the | |
| 02:07 | terminal side , this is this is it right here | |
| 02:09 | , the smallest acute angle between that and the X | |
| 02:12 | axis . So if the angle is really 30 degrees | |
| 02:14 | feta , then fate of prime , which is the | |
| 02:17 | reference angle is also 30 degrees because it's just the | |
| 02:20 | angle to the X axis . That's all it is | |
| 02:22 | . What about 45 degrees If this is the angle | |
| 02:24 | theta , what is fate of prime ? Well fed | |
| 02:26 | . A prime is the same thing . It's the | |
| 02:28 | it's the 45 degrees between it and the X axis | |
| 02:30 | 60 degrees . Same thing . The reference angle here | |
| 02:33 | is between that and the X axis . Now let's | |
| 02:35 | go way over , let's say here to 135 degrees | |
| 02:38 | . You all know that ? This is like the | |
| 02:40 | mirror image of the 45 degree line over here . | |
| 02:43 | Right . So what is the angle here to ? | |
| 02:45 | The angle is measured all the way to here is | |
| 02:48 | what it says , 135 degrees . Now , what | |
| 02:50 | if I ask you what is the reference angle over | |
| 02:53 | there ? The reference angle remember is the smallest angle | |
| 02:56 | which is acute between the terminal side in the X | |
| 02:59 | . Axis . So here's the angle 1 35 . | |
| 03:03 | The angle from here to this X axis over here | |
| 03:06 | is 135 degrees . But the reference angle is asking | |
| 03:09 | me what is the smallest angle ? The acute angle | |
| 03:11 | between this and the nearest X axis . This over | |
| 03:14 | here is the negative X axis . It is the | |
| 03:16 | nearest X axis from it . And so the terminal | |
| 03:19 | angle I'm sorry . The reference angle will be 45 | |
| 03:21 | degrees . If you get a protractor out and stick | |
| 03:24 | it here and measure this angle from here to here | |
| 03:26 | . You will measure 45 degrees . So here the | |
| 03:29 | angle theta is 1 35 . But the reference angle | |
| 03:32 | here is 45 . It's just the angle from wherever | |
| 03:36 | you're at to the X axis . Take away any | |
| 03:39 | signs , no negative numbers . Just what is the | |
| 03:41 | angle between that place where you're at on the unit | |
| 03:44 | circle and the nearest X axis that you have access | |
| 03:47 | to ? So let's just go through rapid fire , | |
| 03:49 | right , We're just gonna go real fast around the | |
| 03:51 | unit circle and then we'll do a bunch of problems | |
| 03:53 | to write them down . What's the reference angle here | |
| 03:55 | ? This is 150 degrees as measured from here . | |
| 03:58 | The reference angle is the smallest angle between this and | |
| 04:00 | the X axis . That's 30 degrees . So the | |
| 04:03 | reference angle between here and here is 30 degrees . | |
| 04:05 | What is the reference angle of 1 21 20 degrees | |
| 04:08 | Is here , the reference angle here is this is | |
| 04:10 | 60 degrees . You can see from symmetry that this | |
| 04:12 | is kind of like the 60 degree line over here | |
| 04:15 | and this is the 45 degree over here and this | |
| 04:17 | is the 30 degree . Of course it's on the | |
| 04:19 | other side . So you could kind of say it's | |
| 04:21 | negative . But for for reference angles we don't care | |
| 04:24 | about signs . I just want to know if I | |
| 04:26 | get a protractor out how many degrees is now , | |
| 04:29 | if you go down here , it's the same thing | |
| 04:30 | . This angle is 210 degrees . That means the | |
| 04:33 | angle all the way measure from here is to 10 | |
| 04:36 | , but the reference angle fate of prime is just | |
| 04:38 | the angle to the nearest X axis . This is | |
| 04:41 | 30 degrees . So the reference angle is 30 degrees | |
| 04:43 | . The reference angle between this and the axis is | |
| 04:46 | 45 degrees . The reference angle between 2 40 over | |
| 04:49 | here is the nearest x axis , which is 240 | |
| 04:52 | degrees . I'm sorry , which is 60 degrees . | |
| 04:55 | Uh You could say the reference angle here is 90 | |
| 04:57 | degrees . When you get over here , the reference | |
| 05:00 | angle no longer goes to this axis , it goes | |
| 05:02 | to the nearest X . Axis , right ? Because | |
| 05:04 | it has to be an acute angle . The reference | |
| 05:06 | angles the smallest angle which is acute . So from | |
| 05:09 | this to the nearest axis is 60 degrees and this | |
| 05:12 | will be 45 degrees and this will be 30 degrees | |
| 05:15 | . You might say . Why do we care about | |
| 05:16 | reference angles ? Because don't we care about the actual | |
| 05:19 | position of the angle ? Well , the answer is | |
| 05:21 | kind of yes . But also reference angles help us | |
| 05:24 | very quickly calculate the sign and the coastline of numbers | |
| 05:27 | anywhere around the unit circle . I'm actually gonna get | |
| 05:30 | to that a little bit later in this lesson for | |
| 05:32 | now . I want you to be able to know | |
| 05:33 | reference angle of this . 30 degrees reference angle of | |
| 05:36 | this . 30 degrees reference angle of this . 30 | |
| 05:39 | degrees reference angle of this . 30 degrees reference angle | |
| 05:42 | of this . 45 degrees reference angle of this . | |
| 05:44 | 45 degrees reference here . 45 . Reference here 45 | |
| 05:48 | . I can go on and on reference here , | |
| 05:49 | 60 . Reference here , 60 . Reference here 60 | |
| 05:53 | . It's just the smallest angle to the nearest X | |
| 05:56 | axis that you have access to . Now . Having | |
| 05:59 | talked through it . These problems should be quite simple | |
| 06:01 | , but I still want to do them because we | |
| 06:03 | need to write something down . So if we have | |
| 06:06 | an X . Y axis here and I tell you | |
| 06:09 | uh here is an angle Like this and I say | |
| 06:14 | , Hey , this angle is Feta and I tell | |
| 06:16 | you to is 60°. . I'm going to ask you | |
| 06:21 | a question what a state of prime equal data . | |
| 06:23 | Prime is the reference angle . So you just look | |
| 06:25 | up here and say , well if this is 60 | |
| 06:26 | degrees then the distance between this to the nearest access | |
| 06:29 | is actually 60 degrees . So in this case in | |
| 06:32 | quadrant one , the reference angle is the same as | |
| 06:35 | the actual measurement of the regular angle in quadrant one | |
| 06:38 | and the other quadrants , then they can be different | |
| 06:41 | . Okay , what if I have over here , | |
| 06:45 | switch things up again a little bit used . Purple | |
| 06:48 | . Let's say over here , I have an angle | |
| 06:52 | . And actually just for completeness , I'm gonna say | |
| 06:55 | since 60 degrees is the angle then fate of prime | |
| 06:58 | is also measured here , which is , I'll label | |
| 07:01 | it on the diagram now over here let's say that | |
| 07:04 | the angle is measured again as it always is from | |
| 07:06 | here . This is angle feta and say the prime | |
| 07:11 | is measured from this position to the nearest X . | |
| 07:15 | Axis . Fate of prime just like this . So | |
| 07:18 | what would if I give you that feta is 150 | |
| 07:23 | degrees ? How do you find that the prime ? | |
| 07:26 | Well , we did it on the diagram over there | |
| 07:28 | because we we kind of are familiar with those diagonal | |
| 07:31 | lines . We know how far away they are from | |
| 07:33 | the positive X . Axis . So we kind of | |
| 07:35 | know on the other side how far away they are | |
| 07:37 | as well . But the real way to do it | |
| 07:39 | is you would say , well we know this black | |
| 07:41 | line is 180 As measured from this direction and we | |
| 07:46 | know that this purple angle is 150 . So really | |
| 07:50 | fate of prime should be 180° -150°. . And so | |
| 07:55 | fate of prime should be 30°. . And that's exactly | |
| 07:59 | what you're doing in your mind . When we go | |
| 08:00 | over there and we look at it were saying how | |
| 08:02 | many degrees is it from here to here ? But | |
| 08:04 | we know that if this is 1 50 this is | |
| 08:06 | 1 80 the difference between them must be 30 degrees | |
| 08:09 | . So what you're really gonna do to find the | |
| 08:11 | reference angle , if it's obvious , you'll just kind | |
| 08:14 | of know what it is . But to do math | |
| 08:16 | , do it is you're going to essentially have to | |
| 08:17 | add or subtract something to figure out what that that | |
| 08:21 | angle is . No matter if it's on top on | |
| 08:23 | bottom negative positive , we don't care about any signs | |
| 08:26 | for the reference angle . We just want to know | |
| 08:28 | how many degrees . If I took a protractor there | |
| 08:30 | to measure it would be All right . So let's | |
| 08:34 | do one in a different quadrant just to get some | |
| 08:37 | practice with something different . Let's say we have an | |
| 08:39 | angle over here in quadrant , number three looks something | |
| 08:43 | like this . And that angle is measured from the | |
| 08:46 | positive X axis . Like it always is data like | |
| 08:48 | this . And I tell you that data is 300 | |
| 08:52 | degrees . And I'll ask you what is data prime | |
| 08:57 | . Well , fate of prime is going to be | |
| 08:59 | the distance or the angle from this terminal line to | |
| 09:01 | the nearest X axis , which is this one . | |
| 09:03 | You're not gonna measure it over here . That's farther | |
| 09:05 | away . It's always going to be an acute angle | |
| 09:08 | . So how do you figure it out ? I | |
| 09:09 | mean , you know that if you go all the | |
| 09:11 | way around this is this black line is 360°. . | |
| 09:15 | So then you take the 360 -300 and you'll get | |
| 09:19 | that fate of prime is going to be equal to | |
| 09:20 | 60° and that's what it is . And you should | |
| 09:23 | do a sanity check just to make sure that this | |
| 09:26 | distance , if it really if this is to 70 | |
| 09:28 | here , the vertical line 300 is here . You | |
| 09:31 | get 60 more degrees to get to 360 . It | |
| 09:33 | makes sense that that's what it is . So oftentimes | |
| 09:36 | when you're solving problems like this , especially later when | |
| 09:39 | we're doing the trig functions , the signs and the | |
| 09:41 | coastlines , you're not really calculating the uh the reference | |
| 09:45 | angle . You're not getting a calculator out . You | |
| 09:47 | just know where it is on the xy plane and | |
| 09:49 | you kind of know from experience how many degrees it | |
| 09:52 | is . You're just subtracting uh some numbers sometimes you're | |
| 09:54 | subtracting or adding 3 60 . Sometimes you're subtracting 180 | |
| 09:59 | basically , it just depends . Is it closer to | |
| 10:01 | this black line over here or is it closer to | |
| 10:03 | this black line that's going to determine what you add | |
| 10:05 | or subtract ? All right . Let's take a look | |
| 10:09 | at a different one . Let's say we have Um | |
| 10:13 | and something over here in quadrant three like this . | |
| 10:18 | So , the angle is fatal here . And we're | |
| 10:20 | saying that data is 225 degrees . This is one | |
| 10:24 | of the ones we actually did over there . So | |
| 10:26 | if the theta angle is measured like this , what | |
| 10:29 | How would you measure the reference angle ? It's just | |
| 10:31 | the angle from this to the nearest X axis . | |
| 10:33 | Label the state of prime . So what do you | |
| 10:36 | do ? Well , if it's just 225 the nearest | |
| 10:39 | X axis is this ? We know this is 1 | |
| 10:41 | 80 . So you either add or subtract 1 80 | |
| 10:44 | but you're not gonna add . That's gonna get a | |
| 10:45 | really big number . You subtract 1 80 so 225 | |
| 10:49 | minus 180 data prime , you're going to get 45 | |
| 10:54 | degrees and that's what it is and that makes sense | |
| 10:57 | because you know that to 25 is one of the | |
| 11:00 | you know this is 45 this is 1 35 this | |
| 11:03 | is 2 25 and then you kind of know those | |
| 11:05 | angles so this one you know is that 45 degree | |
| 11:07 | angle from the access ? So you kind of know | |
| 11:09 | it in the back of your mind . Um But | |
| 11:11 | uh you know showing math and subtracting and adding is | |
| 11:16 | a little trickier . Now these were reference angles when | |
| 11:20 | theta was positive , right ? Data was positive in | |
| 11:23 | all these cases measured from the X axis . What | |
| 11:25 | happens if the original angles negative doesn't make it harder | |
| 11:28 | or not ? Just remember what you're doing ? Is | |
| 11:31 | it no matter what you actually mathematically have to do | |
| 11:34 | , you just want to know what the angle is | |
| 11:35 | between wherever you're at in the X axis . So | |
| 11:38 | let's say that we have something like this over here | |
| 11:43 | , we have an angle like this and it's measurement | |
| 11:46 | is we call feta and then theta is defined to | |
| 11:49 | be negative 210°. . So actually I made a mistake | |
| 11:53 | . Let me kind of subtract that shouldn't have drawn | |
| 11:55 | that in this problem . Sorry about that . What | |
| 11:58 | I'm really trying to say is the the the terminal | |
| 12:01 | side is there but really it's measured here now this | |
| 12:03 | is called data and data happens to be negative . | |
| 12:06 | So when you think about it , I mean it | |
| 12:08 | makes sense if you go this direction , this is | |
| 12:10 | negative 1 80 so a little beyond negative 1 80 | |
| 12:13 | is up there . Um Now again , I think | |
| 12:16 | you probably already know the answer just by looking at | |
| 12:18 | it . But the terminal or the reference angle would | |
| 12:20 | be this measurement right here . Fada crime . So | |
| 12:22 | what do you do ? A lot of students don't | |
| 12:24 | know should I add or subtract ? Right . Well | |
| 12:27 | , ultimately , no matter if it's negative measure or | |
| 12:30 | positive measure , all you want to know is how | |
| 12:33 | many degrees is this ? So yeah , there's a | |
| 12:36 | way to do it with negative negative numbers . But | |
| 12:38 | ultimately , what you need to do is say , | |
| 12:40 | well , okay , it's gonna be this 210 minus | |
| 12:43 | 180 . Those are the two relevant numbers . If | |
| 12:46 | I subtract them , what I'm going to get is | |
| 12:48 | 30 degrees , which makes sense from this number because | |
| 12:51 | if I go in the negative direction and land here | |
| 12:53 | , it's negative 1 80 . But I go another | |
| 12:56 | 30 degrees up to get to negative 2 10 . | |
| 12:58 | So it is a 30 degree angle and you can | |
| 13:00 | convince yourself of that . But still sometimes people aren't | |
| 13:03 | sure why why did I take the negative away here | |
| 13:05 | ? Why did I do this ? Well the the | |
| 13:07 | answer is the angle measure is negative to 10 . | |
| 13:10 | And then the line over here is negative 1 80 | |
| 13:13 | . So you could subtract them negative to 10 minus | |
| 13:17 | a negative 80 which would be plus 80 . And | |
| 13:19 | you would get negative 30 degrees as the angle . | |
| 13:22 | But we strip away all signs . We don't care | |
| 13:24 | if I'm measuring going up or coming down . The | |
| 13:26 | negative sign tells me which way I'm measuring going up | |
| 13:28 | or coming down but I don't care about that . | |
| 13:30 | I just want to know what the angle is . | |
| 13:33 | Is a positive number . So you would strip away | |
| 13:35 | the sign anyway I find it uh much easier just | |
| 13:38 | to take the numbers and subtract and get a number | |
| 13:40 | out of it . Alright , next problem . What | |
| 13:43 | if we have something like this and we have something | |
| 13:48 | over here looks like it's in quadrant one and the | |
| 13:51 | angle here is not gonna be negative but it goes | |
| 13:53 | around all the way around like this , this is | |
| 13:56 | data and we say that theta is equal to 420 | |
| 14:00 | degrees . What is the reference angle , first of | |
| 14:02 | all on the diagram , what is the reference angle | |
| 14:05 | ? It's the smallest angle to the nearest X . | |
| 14:07 | Axis . So something like this , this is the | |
| 14:09 | angle that I want right here . Um You can | |
| 14:12 | draw the arrow down here , you can draw the | |
| 14:14 | arrow appear . Doesn't really matter too much . But | |
| 14:16 | anyway this is state of prime , that's what I'm | |
| 14:17 | after Now . It's a big angle , it goes | |
| 14:21 | all the way around 360 . Then it goes some | |
| 14:23 | distance up . I want to know what this extra | |
| 14:25 | distances up . That's gonna be a beta prime . | |
| 14:27 | All right . So , since it goes all the | |
| 14:30 | way around and you're measuring to the black line , | |
| 14:32 | what you do is you say data prime is 420 | |
| 14:36 | -360 . Data prime is going to be there is | |
| 14:40 | going to be 60° and then you look and see | |
| 14:42 | , does it make sense ? Right ? If I | |
| 14:44 | go all the way around 23 63 60 plus 60 | |
| 14:48 | more is going to be going up to 4 20 | |
| 14:51 | which is where it is . So all I'm after | |
| 14:53 | is what is this angle measure here ? So sometimes | |
| 14:56 | you subtract 3 60 . Sometimes you subtract 1 80 | |
| 14:58 | . It just depends on if the terminal side is | |
| 15:00 | over here somewhere or if the terminal side is over | |
| 15:03 | here somewhere , strip away all signs , strip away | |
| 15:06 | negative numbers and all of that . And you're going | |
| 15:08 | to get the reference angle . Now we get to | |
| 15:11 | the the important part . Why are we calculating a | |
| 15:13 | reference angle ? Why do we care ? Here's the | |
| 15:16 | reason why . Let me do a little bit of | |
| 15:17 | a lecture and then we're gonna solve a couple of | |
| 15:19 | quick problems . Mhm . In the past we have | |
| 15:23 | learned how to find the sign and the co sign | |
| 15:26 | of these numbers over here . These angles . Right | |
| 15:28 | ? So let's pick one that we know really well | |
| 15:30 | , sign of 30 is one half this number right | |
| 15:33 | here . This is the ex the co sign value | |
| 15:35 | and this is the sign value . So we know | |
| 15:36 | that sign of 30 is one half . That means | |
| 15:39 | that the projection over here is one half right . | |
| 15:42 | And we also know that the co sign of this | |
| 15:45 | number is the square root of 3/2 . That's the | |
| 15:48 | value if you project it down here , remember I | |
| 15:50 | told you a long time ago that the sign is | |
| 15:53 | the projection onto the Y axis . Sign goes with | |
| 15:56 | why ? And the co sign is the projection onto | |
| 15:59 | the X axis . It's just as if I shine | |
| 16:01 | a light on this hypotenuse and it cast a shadow | |
| 16:04 | this much is going to be one half . And | |
| 16:06 | if I shine a light down and it cuts and | |
| 16:08 | makes a shadow here this distance , since it's a | |
| 16:11 | unit circle with a radius of one will be squared | |
| 16:13 | of 3/2 . It's about 10.866 or so . And | |
| 16:17 | so this is the shadow that would cast right there | |
| 16:19 | in quadrant one , all of the projections are positive | |
| 16:23 | because the X axis is positive and the Y axis | |
| 16:26 | is positive but in the other quadrants X and Y | |
| 16:29 | are not positive . So when the other quadrants , | |
| 16:32 | the sign can give you a projection that's negative . | |
| 16:35 | And the co sign can also give you a projection | |
| 16:37 | is negative . Let me show you what I mean | |
| 16:40 | , we all know that this is 150°, , we | |
| 16:43 | know that , right ? So If I say what | |
| 16:47 | is the sign of 150° since it looks exactly like | |
| 16:50 | the mirror image of this one , you know that | |
| 16:53 | if I shine a projection like this and it cast | |
| 16:56 | a shadow right here , it's gonna be the exact | |
| 16:58 | same value if I cast it from here onto the | |
| 17:00 | 30 degree because of symmetry . So it's gonna be | |
| 17:02 | one half . So we now know that the sign | |
| 17:05 | of this 150 is going to go land on the | |
| 17:07 | same place , the positive y axis over here , | |
| 17:09 | It's gonna give me the same sign . The sign | |
| 17:11 | of 30 is the same exact thing , is the | |
| 17:14 | sign of 1 50 because the projection on the Y | |
| 17:17 | axis is exactly the same for both of those angles | |
| 17:20 | . So when you're looking at angles in quadrant one | |
| 17:22 | and two , the signs of those angles are always | |
| 17:25 | the same . The sign of 120 is exactly the | |
| 17:28 | same as the sign of 60 because if I take | |
| 17:31 | a projection over here and I take a projection over | |
| 17:33 | here , the sign which is the projection online is | |
| 17:36 | exactly the same for those . But let's go and | |
| 17:38 | examine this one and this one again , the co | |
| 17:41 | sign over here was the projection over here , that's | |
| 17:43 | positive values of X . But the projection down here | |
| 17:47 | on the X axis is not positive , these are | |
| 17:49 | all negative values of X . So if I shine | |
| 17:53 | the value down it's gonna cut over here . I | |
| 17:55 | expect to get a negative co sign . That is | |
| 17:59 | the secret sauce of sine and cosine . To figure | |
| 18:02 | out what the sign of a value is . You | |
| 18:04 | figure out where it is um unit circle and you | |
| 18:05 | project it and see where it lands on the Y | |
| 18:08 | axis . If the projection is anywhere up here , | |
| 18:11 | the sign is a positive number somewhere between zero and | |
| 18:14 | one . If you are down here and you do | |
| 18:17 | a projection on the Y axis then the y axis | |
| 18:20 | is negative . And you'll always get negative signs for | |
| 18:22 | all of these angles because all of their projections will | |
| 18:25 | all be a negative . Why likewise , anybody over | |
| 18:30 | on the right hand side the positive X axis . | |
| 18:33 | Any of these angles will give me projections that are | |
| 18:36 | positive , which means the coastlines will be positive . | |
| 18:38 | And over here the coastlines will always be negative for | |
| 18:41 | all of these angles all the way over here because | |
| 18:43 | any projection from down below or up above is gonna | |
| 18:46 | land over here . So the secret sauce of figuring | |
| 18:49 | out what the sign or the coastline is of any | |
| 18:51 | of these angles around the unit circle is to figure | |
| 18:54 | out the reference angle first . Because if I know | |
| 18:57 | the reference angle here is 30° And I already know | |
| 19:01 | what the sign and co sign of 30° is from | |
| 19:03 | quadrant one , then I know what the number of | |
| 19:05 | the sign in the coastline is . The numbers are | |
| 19:07 | all written in red . Around here , around the | |
| 19:09 | unit circle one half . The sign is one half | |
| 19:12 | of 30 degrees and the coastline is square to 3/2 | |
| 19:14 | . But in the other quadrants , the sine and | |
| 19:19 | cosine will have the same numbers but different signs . | |
| 19:21 | So we get , for instance , say what is | |
| 19:23 | the co sign of 1 50 ? Well the coastline | |
| 19:25 | 1 50 is the projection down here . We know | |
| 19:27 | the coastline of 30 is square to 3/2 . So | |
| 19:30 | we know the co sign of 1 50 must be | |
| 19:33 | the same number because of cemetery , it's square to | |
| 19:36 | 3/2 . But this projects down here to negative square | |
| 19:38 | root of 3/2 . So the co sign of 1 | |
| 19:41 | 50 is negative square to three . Over to the | |
| 19:43 | co sign of 30 is positive square to 3/2 . | |
| 19:46 | And so that is basically what we're gonna do . | |
| 19:48 | We're gonna walk around the unit circle and we're going | |
| 19:50 | to figure out the sign and the co sign first | |
| 19:52 | by figuring out what is the reference angle , what | |
| 19:54 | is the reference angle ? Once we know the reference | |
| 19:57 | angle we know what the sine and cosine numbers are | |
| 19:59 | in quadrant one . But then we'll just take those | |
| 20:01 | numbers and we'll put signs on them , either negative | |
| 20:03 | or positive signs to figure out what actually is going | |
| 20:06 | on . So with that understanding let's go and try | |
| 20:12 | to make some sense of the next problem . This | |
| 20:16 | is a great set of problems coming up . They're | |
| 20:18 | extremely important . Okay , so I want to write | |
| 20:23 | Co sign of 200° as a function of some acute | |
| 20:32 | reference angle . Now this problem looks really hard but | |
| 20:36 | actually we know everything needed To do this . We | |
| 20:39 | want to find right the co sign of 200 . | |
| 20:41 | I notice that 200 is not an angle that we | |
| 20:43 | know so well in the unit circle , But still | |
| 20:46 | , once we know the coastline of 200 , how | |
| 20:48 | do we write it in terms of a co sign | |
| 20:50 | of some other reference angle . The very first thing | |
| 20:53 | you need to do is draw a graph and figure | |
| 20:55 | out where on the unit circle this 200 is so | |
| 20:58 | that you can visualize it . I'm telling you right | |
| 21:00 | now in the beginning , if you're not drawing pictures | |
| 21:02 | are going to get something wrong . So you know | |
| 21:04 | that this is zero , this is 90 this is | |
| 21:07 | 1 80 this is 2 70 . So one eighties | |
| 21:09 | here . So a little bit beyond that is 200 | |
| 21:11 | degrees . So let's draw our terminal angle over here | |
| 21:15 | like this . So we know that this is 180 | |
| 21:18 | Right ? And we know that this angle right here | |
| 21:21 | is measured as 200 degrees . And because it's 200 | |
| 21:26 | degrees we know something really critical . What is the | |
| 21:29 | reference angle here ? We just studied what the reference | |
| 21:31 | angle is . The angle to the positive X axis | |
| 21:34 | . This or not ? The positive X axis . | |
| 21:35 | The nearest X axis . This angle right here , | |
| 21:39 | we call it fate of prime . The reference angle | |
| 21:41 | is actually 20 degrees . How do I know that | |
| 21:44 | ? Because this thing is 200 degrees and the nearest | |
| 21:46 | X axis here is 1 80 . So we subtract | |
| 21:48 | them , we get 20 . This if you get | |
| 21:50 | a protractor you're only gonna measure a 20° angle . | |
| 21:53 | Okay , now , here's the critical piece of information | |
| 21:57 | . The projection of co sign 200° is going to | |
| 22:04 | be negative . Remember ? Cosign projects to the X | |
| 22:08 | axis . If this is the angle if we shine | |
| 22:12 | a light and project it to the nearest X axis | |
| 22:14 | , the X values are all negative over here . | |
| 22:16 | The X values are positive over here , but the | |
| 22:18 | X values are negative over here . So this projection | |
| 22:21 | is going to project onto the negative X axis . | |
| 22:24 | That means the co sign of that angle will be | |
| 22:27 | negative . All you do is you look at where | |
| 22:29 | it's at , you projected on the X axis for | |
| 22:31 | a cosign question and if it's in the negative X | |
| 22:33 | axis is going to be a negative co sign . | |
| 22:35 | So the projection of this angle when you're taking the | |
| 22:38 | co sign is going to be in the negative access | |
| 22:41 | there because it's projecting up to this guy right here | |
| 22:45 | . All right then . So the question then is | |
| 22:48 | Uh what I want to do is write co sign | |
| 22:51 | of 200 degrees as a function of the acute reference | |
| 22:57 | angle . We now know the reference angle is 20 | |
| 22:59 | degrees . So what we're basically saying is that co | |
| 23:02 | sign of 200 degrees is exactly the same as negative | |
| 23:06 | co sign . Whoops messed up already . Sorry about | |
| 23:08 | that negative co sign of 20 degrees . Now let's | |
| 23:13 | let's break that down for a second and make sure | |
| 23:15 | you understand . Let me actually get this out of | |
| 23:17 | the way because I'm going to need to draw something | |
| 23:19 | up here . This is still a 200 degree angle | |
| 23:22 | , Right ? It's negative of coastline of 20 . | |
| 23:24 | Why ? Because what does a 20° angle look like | |
| 23:27 | that look like a 20° angle ? If I were | |
| 23:29 | to draw it in quadrant one is gonna look like | |
| 23:31 | this this angle is a 20 degree angle in quadrant | |
| 23:36 | one . So if I take like I know that | |
| 23:39 | I don't know in my mind what co sign of | |
| 23:40 | 20 is . So it's not one of the special | |
| 23:42 | angles but I know that it has some value right | |
| 23:45 | ? I know that I can project it onto the | |
| 23:46 | X axis . If I go figure out what this | |
| 23:48 | projection is , it's gonna in the in the first | |
| 23:51 | quadrant it'll give me some number and whatever that number | |
| 23:54 | is with the exact same absolute value as the projection | |
| 23:57 | over here because the reference angle , the 20 degree | |
| 24:00 | reference angle that's over here , the projection on this | |
| 24:03 | negative X axis is going to be the same number | |
| 24:05 | as the projection over here , it's just this one | |
| 24:08 | will be negative and this one will be positive . | |
| 24:10 | So what you do when you have angles on the | |
| 24:12 | other side of the unit circle is you figure out | |
| 24:14 | the reference angle first , then you take either the | |
| 24:17 | sign or the coastline of it and you get the | |
| 24:19 | value and then you slap a sign on it depending | |
| 24:23 | on what quadrant it's in . In this case we | |
| 24:25 | go and we say this is a 20 degree reference | |
| 24:27 | angle . So if it were over here , whatever | |
| 24:29 | the value of this projection to the co sign to | |
| 24:32 | the X axis is , whatever it is , we're | |
| 24:34 | gonna stick a negative on it because the real projection | |
| 24:36 | is over here , projecting onto this axis over here | |
| 24:39 | critically critically important . I encourage you to work the | |
| 24:45 | rest of these problems with me . And then as | |
| 24:47 | we go on around the unit circle , we're gonna | |
| 24:48 | do a lot more of this as well . Let's | |
| 24:51 | do another one . Uh We want to write as | |
| 24:53 | a function of an acute angle here . Uh of | |
| 24:56 | the acute reference single . What about the sign of | |
| 24:59 | negative 17 degrees ? The sign of negative 17 degrees | |
| 25:04 | . First things first , we must figure out where | |
| 25:07 | in the in the unit circle is this angle ? | |
| 25:10 | What is the reference angle ? Well , it's negative | |
| 25:13 | 17 degrees . That means it goes down below the | |
| 25:16 | X . Axis just a little bit 17 degrees . | |
| 25:19 | Right ? So here this angle here is negative 17 | |
| 25:24 | degrees . That's what that angle is . Now . | |
| 25:26 | What is the reference angle ? The reference angle is | |
| 25:29 | the positive acute angle from wherever you are to the | |
| 25:32 | X . Axis . So the actual reference angle so | |
| 25:35 | we'll say data is negative 17 degrees . But Fate | |
| 25:39 | a prime is actually positive 17 , whoops , positive | |
| 25:43 | 17 degrees . It's just the absolute value of that | |
| 25:46 | because it's just , how many degrees are you like | |
| 25:48 | positive negative throw it away . The reference angle is | |
| 25:50 | 17°. . So if I wanted to draw this reference | |
| 25:53 | angle it would be something like this . Data . | |
| 25:57 | Prime is 17°. . So basically you take any value | |
| 26:02 | anywhere in the circle you find the reference angle and | |
| 26:04 | then you think about what does the reference angle gonna | |
| 26:07 | do in quadrant one . That's going to give me | |
| 26:09 | the value of the sign of the coastline . Whatever | |
| 26:12 | I'm talking about then to find the value of what | |
| 26:14 | I'm really looking for . I just put a sign | |
| 26:16 | like a negative or positive sign on it . So | |
| 26:19 | in this case I want to write this sign of | |
| 26:22 | -17° is going to be uh what is it going | |
| 26:28 | to be ? It's going to be negative sign of | |
| 26:32 | positive 17 degrees . In other words , this 17 | |
| 26:35 | degree angle that exists up there , it's going to | |
| 26:37 | give me some projection onto this case . I'm gesturing | |
| 26:41 | to the X axis , that's actually a sign . | |
| 26:43 | So we're finding the projection onto the Y axis here | |
| 26:46 | . But this blue line is going to give me | |
| 26:47 | a positive projection . Whatever the projection is . Whatever | |
| 26:51 | answer I get , I slap a negative sign on | |
| 26:53 | it and that's gonna be the projection of this one | |
| 26:55 | because this one here is really being projected onto the | |
| 26:58 | negative axis here . So we're saying that the sign | |
| 27:02 | of -17° which is the projection of this guy onto | |
| 27:06 | this axis will give us a negative value . What | |
| 27:08 | will it be ? It will be the negative of | |
| 27:10 | whatever the sign is of that positive angle , because | |
| 27:13 | the projection here gives us a positive value . We | |
| 27:16 | slap a negative on it and that's gonna give us | |
| 27:18 | the projection of what we really are after . So | |
| 27:20 | these problems seem crazy , weird , but they're actually | |
| 27:23 | exactly what you're going to do when you find you | |
| 27:26 | know what is the sign of 100 and 50 degrees | |
| 27:28 | on the other side of the unit circle . You're | |
| 27:30 | going to think about how far is it to the | |
| 27:31 | X axis ? Figure out what the sign and co | |
| 27:33 | sign is and then put a negative or a positive | |
| 27:35 | value . Wanted to get the final answer ? Yeah | |
| 27:38 | . All right , let's do another one . Let's | |
| 27:40 | say we have the co sign Of -221 9°. . | |
| 27:46 | Like a weird little angle like this . First of | |
| 27:48 | all , we need to figure out what quadrant this | |
| 27:50 | is in and we always need to draw a picture | |
| 27:51 | to do that properly . So it's negative 2 21 | |
| 27:56 | . So here's negative 1 80 that's negative to 70 | |
| 27:59 | . So somewhere over here , right , Is this | |
| 28:02 | angle like this ? And so we say data is | |
| 28:06 | negative 221 9°. . Something like this . All right | |
| 28:11 | . So then the question is what if we want | |
| 28:14 | to figure out if we want to write this in | |
| 28:16 | terms of the coastline of another angle ? A positive | |
| 28:18 | acute angle . Then the first thing we have to | |
| 28:20 | do is figure out what that positive acute angle is | |
| 28:23 | . What's the reference angle ? This angle right here | |
| 28:25 | is the reference angle . Theta prime . So what | |
| 28:28 | will it be data prime ? Well let's just take | |
| 28:30 | this number 2 to 1.9 minus 180 . And let | |
| 28:36 | me make sure I did my math right here . | |
| 28:38 | 41.9°. . Mhm . That means if I get a | |
| 28:42 | protractor and actually measure this , I get 41.9°. . | |
| 28:46 | All right . Like this . So if I go | |
| 28:48 | from symmetry and take a look at it at an | |
| 28:51 | equivalent angle over here , that's exactly the same thing | |
| 28:54 | . 41.9° like this right measure to the positive X | |
| 28:59 | . Axis . If I figure out whatever the co | |
| 29:01 | sign of this angle is , the actual number I | |
| 29:04 | get will be the exact same number that I would | |
| 29:05 | get over here . But I'd have to stick a | |
| 29:07 | negative sign because this is projecting to the negative part | |
| 29:10 | of the X . Axis over here . So the | |
| 29:12 | bottom line is the co sign of negative to 21.9 | |
| 29:17 | uh is going to be projected Into the negative negative | |
| 29:23 | access . So this projection to the co sign here | |
| 29:25 | is going to give me a negative number . So | |
| 29:27 | because of that the co sign of negative 2-1 9° | |
| 29:32 | is going to be equal to The negative of the | |
| 29:36 | co sign of 41.9° negative of coastline of 41.9°. . | |
| 29:43 | So I encourage you to do this . Get a | |
| 29:44 | calculator put 41.9°. . Make sure your degree mode . | |
| 29:49 | Hit the coastline button and slap a negative on it | |
| 29:52 | and figure out what answer you get . You're gonna | |
| 29:53 | get some number . Then separately put negative 2-1.9 in | |
| 29:58 | the calculator . Hit the coastline button . Those two | |
| 30:00 | numbers will be the same if you're trying to figure | |
| 30:03 | out what the coastline of this negative angle is . | |
| 30:05 | What we always do in trigonometry is we say , | |
| 30:08 | well , first of all , what is the angle | |
| 30:09 | ? Even though I know it's way over here , | |
| 30:11 | what is the angle between this and the positive and | |
| 30:14 | whatever the closest accesses ? |
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