The Easiest Way to Memorize the Trigonometric Unit Circle

[@genevalley5083]

Thank you. After 30 years and no degree from UC Irvine, I understand this better. I opened my original calculus book and I'm going through all of it again now. Your video helps greatly!

[@yamanydadwaria]

👏🏻 great sir

[@chewinggum5550]

Good luck sir!

[@hadeelalansari5421]

good luck!

[@Benevolenttt]

Great sirrr

[@worldshaper1723]

Love to meet you

[@thefunbuns1]

Your visualization method just made this click for me. I'm studying up because I'm going back to school as an ex-dropout and have to take Calc 2, but I'm so rusty. I used to draw the unit circle on all my tests in AP Calc to help but after watching this I could actually do this in my head, I am blown away

[@hemal6437]

After a long 17 years of life thinking that Maths is hard , You Made it A piece of cake , THANK YOU DAVE , For teaching us .

[@youtubeuserdan4017]

More like a slice of pi.

[@ZallyAimro]

no, a piece of pie :D not cake

[@rainerpour2501]

I wish math teachers used these methods around the world; we would still have math haters; but those who really wanted to learn would have a much easier time.

[@pinklady7184]

Thank goodness for KZbin.

[@mustafenur3242]

your making things complicated twin which highly affected our confidence level

[@EhsanulHaque-g8g]

you make my college life easier. i was looking for this video, I watched every video related to this topic. I understand the other concepts.

[@DoIt-fg7rp]

College!! Wr do learn this in 9th grade

[@TheKids422]

What degree are you pursuing?

[@takemehomecountryroads657]

Best explanation ever. This has helped me a lot with Simple Harmonic Motions and Theoretical mechanics.

[@blackplank2580]

You only need to remember the pattern for square roots is √2/2,√2/2. √3/2,1/2. 1/2,√3/2

[@nicholasssin]

Wrong order but yes

[@michaelstewart1526]

"just stare at this, until it makes sense"....i love you

[@belgiumball2308]

Ahhhhh Professor Dave saves trigonometry students again

[@besisis1308]

thank uuuuu. it has helped a lot in geometry a few years ago and pre-calculus this year!!! keep up the great work!

[@mahendraify]

Very nice..!!! Ur voice is so mesmerizing.

[@TalkedSniper322]

Was struggling a lot on this for Math 112... thank you so much.

[@georgesadler7830]

Professor Dave , thank you for another excellent video/lecture on The Easiest Way to Memorize the Trigonometric Unit Circle. One of the best ways to learn the Unit Circle is by pattern recognition.

[@StudywithmeinPakistan]

Probably one of the best explaination ever.

[@GeomaniaDasher]

I was just thinking about this today, how the standard triangles easily define the unit circle. Sure enough you had already thought to make a video explaining it so simple! Wish more people knew about your videos because they save my life

[@BrunoPeitl84]

I really enjoyed your tutorials and learned a lot from them. But if you allow me some constructive criticism, I've missed a few things in these tutorials on trigonometry. You could have demonstrated, for example, where the trigonometric functions of the angles of 30, 45 and 60 degrees came from. (You gave some clues in geometry tutorials, but there were still some gaps.) You could have explained why the radius of the unit circle is 1 (because it equals the length of the circumference [2.pi.r] and the angle of 360 degrees in radians [2.pi.rad].) And in this tutorial specifically, you could have explained why the x and y coordinates correspond to the sine and cosine of the angle and why the tangent is equal to sine over cosine.

[@RedRahiq]

Check earlier videos.

[@jcpmac1]

Prof. Dave always gives me the impression of a man driven by a sense of purpose, and is determined not to have it impeded even momentarily by having to give his attention to those wearisome whimpered questions from the dullards at the back of the class.

[@isaachall8600]

Thanks! I'm an engineering major and had an trig exam 2 weeks ago. You explained it better than my class!

[@TheKids422]

You're an engineering major, what discipline?

[@rehoboth_farm]

There is no reason to memorize the unit circle. The easy way to produce the unit circle is to divide a circle into 24 equal 15 degree graduations. Number these points from zero Pi to 2 Pi 0,1,2,3... then take those numbers and multiply them by Pi/12. You will have 8 extra locations on the your unit circle that you can use or throw away. The same method can be used to give you any number of equal graduations that is an even number. Just change the denominator to half that number.

[@J_Duba]

7:14 "Just stare at this until it makes sense" Weirdly enough, this worked

[@Pathshala260]

Thanks bro , I am from India a tuition teacher you helped me to knowing better way

[@nicolasbourbaki314]

this video should be compulsorily watched at schools.

[@blueblizzard7551]

ah, i kneel before you again professor dave. asking to be blessed for this trig unit we’re doing in quarantine.

[@Legendfound-x8e]

Your video is the best in explaning the unit angles.

[@arlenefeliciano8066]

Very well explained, love it. And good voice tone. Congratulations!

[@AnwarHossain-rq2ft]

Thanks.for your good teaching.

[@amiyakumarchowdhury5283]

Bangladeshi?

[@conradfung5537]

I learn't the quadrant signs as "CAST" , fourth quadrant = "C" for cos, first quadrant = "A" for all signs, second quadrant = "S" for sin, and third quadrant = "T" for tan.

[@ninnibugge2755]

You just saved me for my math exam in an hour! THANK YOU!!

[@VimbaiKugara]

very helpful thank you Professor Dave

[@janah9789]

Thank you so much for this. I have a quiz to write 15 sin , cos, and tan points from radians in under 2 minutes and never understood how to memorize. This cos and sin trick really helped me for this quiz next week.

[@awolgeordie9926]

Excellent lesson Prof Dave!!

[@davidkwon1872]

Who starts the video from the beginning to hear the video intro 5 times.

[@ellangel6139]

🥳😇💃

[@indigofenrir7236]

So this is my understanding: - Right angles have radians xπ/180 where x = degrees. - 30° angles have a radians denominator of 6 and a numerator coprime to 6, times π. They have cos and sin (√3/2, ½) with signs depending on the quadrant. The angles are closer to the x-axis. - 60° angles have a radians denominator of 3 and a numerator coprime to 3, times π. They have cos and sin (½, √3/2) with signs depending on the quadrant. The angles are closer to the y-axis. - 45° angles have a radians denominator of 4 and an odd numerator, times π. They have cos and sin (√2/2, √2/2) with signs depending on the quadrant. Suddenly this doesn't seem quite as intimidating as it was 10 years ago.

[@Islandpulledfromthesea]

The 114 people who disliked failed the math exam.

[@nicholasssin]

I owe my 100 on that test to you 🙏🙏🙏❤

[@ComputerCurry]

Thanks for the intuitive explanation.

[@GielIntven]

Thank you so much! This really helped me a ton!

[@fuelknightmare]

I can't wrap this concept around my head 😭😭😭

[@AustinWoll]

This is fantastic! Thank you!

[@vikrampatil2748]

This was very helpful Thank you Dave

[@yajatyadav3348]

Lol. My autoplay put on the video Why you should never memorize the unit circle after this video ended.

[@ivoryas1696]

Yajat Yadav Wait, why wouldn't you want to memorize it?

[@ivoryas1696]

@GOG REACH Oh, yeah. I saw it and it's actually a pretty sensible argument either way. I've personal gone about half and half because I didn't study it _quite_ enough to memorize it all 😅 Thanks anyway, though.

[@s.hariteja9414]

Sir, can you say me how the coordinates of π/4 is (√2/2,√2/2). As the coordinates must be (1/√2,1/√2)[ cos 45 = 1/√2 , sin 45 = 1/√2 ]

[@Ilikecats5]

This is because we don't like have roots in the denominator, so we multiple the top and bottom by root 2, and that gives us root 2/2

[@keeranmnc1605]

It's called rationalizing the denominator

[@jeanne5067]

Hi, I don't know if anyone will find this. But I got a problem understanding the last exercice. My final answer was 1 but professor dave said it was -1. My process: sin(7pi/2) - 6pi/2 (=>3 full revolution) sin(pi/2) which is equal to the 90° angle. we are searching sin, so the coordinate y which would be 1 for this angle. I don't understand why it's negative... Thanks in advance ^-^.

[@Joshzxc00]

I had the same thought, but if you think it a little you notice: So 6pi/2 means 3pi and you’re left with 1pi/2, so you could say you have 3,5pi. 2pi means one rotation (stating from 0 on the right). Then another 1pi means half the circle and the last half of pi means the point(0, -1) on the circle. In the parenthesis you have (cos, sin) so the sin of that point is -1. Basically you just need to understand how to “rotate” using the given pi. Hope what i wrote makes sense, it’s how i understood it. Good luck!!

[@kellyrojers4202]

@@Joshzxc00 I was thinking the same thing lol

[@jeanne5067]

@@Joshzxc00 Okay I see, I'll try to apply that on other exercises thank you! :D

[@ceegae1670]

Well do it the easier way My process goes: 7pi/2 - 4pi/2 = 3pi/2 (one full rotation since 4pi/2 = 2pi) And 3pi/2 is the point (0, -1) Hope this helps.

[@ampisiades]

It will be ALWAYS Very helpful.Thanks Professor Dave.

[@evryon1810]

This video is actually good

[@janetanwar7573]

I didn't get csc -17pi/4 why did we consider the axis 0 16pi/4 although we considered it 12pi/4 for the previous problem which was tan 14pi/3 ?

[@curtpiazza1688]

Thanx Prof Dave!

[@Jeff_Segal]

I think the only thing you really need to memorise is sqrt3/2. If you know that one revolution = 2pi, and know how to use Pythagoras, the rest are easy to derive on the fly.

[@rajanvk939]

Why there always have to be value 1 for perpendicular side although circle ⭕️ has radius 1 which is hypotenuse. Why hypotenuse was taken 2 for triangle?

[@MxOwl]

So I think I understand everything except for one of the problems in the practice part. Sin(7pi/2), can’t we make 3revs (6pi/2) and then it’s just (pi/2) which would be 1. But the answer says -1?

[@sainteilish778]

did u figure it out? pls explain to me if u did

[@ceegae1670]

​@@sainteilish778 Take 4pi/2 which is 2 revolutions, then you're left with 3pi/2 , which is the point (0, -1) => sin = -1

[@taimulahmedjisan3590]

Love from Bangladesh sir❤🇧🇩🇧🇩

[@ninjapirate123]

That was a great explanation

[@tGoldenPhoenix]

Done.

[@CJCREATIONS001]

happy new year

[@Ilikecats5]

I got confused at 7:27 I don't understand where the 12pi/3 came from tan(14pi/3). Maybe I'm just dumb at this but could someone explain it simpler for me

[@maggiejohnson9232]

3/3 = 360° or 1 rev. So, 12/3 = 4(360°) or 4 rev. Then, 14/3 - 12/3 = 2/3 * Ignore the first 4 revolutions and focus on the remaining new revolution which is 2/3 That's why, 14π/3 is equal to 2π/3

[@horizonbrave1533]

But what about all the points inbetween these selected ones?

[@alalla2173]

Very good

[@nextpage9680]

Thank youu sir💕

[@gregl2249]

There is absolutely no reason to have to memorize this except teacher exerting their dominance

[@tamizhtamizh412]

Love you bro 😇

[@shusin6701]

7:22 start here difficult to understand

[@iloveferrets5859]

Bro when you said "just take my word for it sin30 is 1/2" I laughed harder than I should have

[@flaneurable]

Alright, so what do you do with the co-ordinates angle 30 degrees?

[@coryintheboof8730]

I know pi/2. Is supposed to litterally ration out to half of pi. But intuitively I dont look at it that way. Like with pizza. If you have 1/2 slices. You think. Well. How many times can the nominator fit in the denominator. And thats twice. To make a whole, in this case, a whole pizza. So if 2/2 is the entire pizza. 1/2 is half of that. But with pie/2. When you are at a quarter of the circle. It is 3.1416/2. And If thats. Half. Then times 2 will get my whole. Of 6.2832/2. Which makes no sense. Help me intuitively think of it right.

[@ProfessorDaveExplains]

Once around the circle is 2 pi. Half around is pi. Quarter of the way around is half pi. Check out the previous tutorial for the definition of the radian, it will help.

[@ProfessorDaveExplains]

Sorry I'm still having a bit of trouble understanding where the confusion lies, it's possible it is with fractions in general, in which case you may want to visit some of my earlier tutorials in the mathematics playlist. You wrote out pi over two pi, which is not the same thing as pi over two, what you wrote would be one half. But either way, you really should start from the beginning of the trigonometry playlist, it should clarify a few things.

[@coryintheboof8730]

Professor Dave Explains i did more math today. I was just misunderstanding how everything fit with each other. Im pretty good at math usually and this was dumb lol. But Im good. Sorry about that.

[@ProfessorDaveExplains]

no problem, we are all here to learn! please check out more of my math tutorials, they will help you.

[@carultch]

@@coryintheboof8730 Consider a 24 inch pizza, which would be a unit circle in units of feet. Measure the length of the crust around the circle, also in units of feet. That is how you determine the angle of each slice in radians. If you measure this giant pizza's total crust length, you will get 6 feet 3.4 inches, which is 6.28 feet. This is 2*pi*radius. When the radius is 1, arc length around the circle is the angle in radians. Don't let it deceive you that a pizza is sometimes called a pie or that pie sounds similar to the name of the letter pi. It is just a coincidence, and is an artifact of the English language's evolution that this happens. In the original Greek, the letter pi sounds like the English word "pea". They pronounce the letter "P" like the English word "pay", when Latin letters are used in modern Greek mathematics.

[@priyalpradhan2777]

Since its the unit circle and the radius is supposed to be 1, the hypotenuse is 1, why are we taking the opposite side as 1??

[@OrchestratedChicanery]

Where did he take the opp side as 1? Please give more details.

[@carultch]

If you draw a right triangle with the point on the unit circle, and the origin, and the X-axis, the hypotenuse will be the same as the radius of 1. The x-coordinate will be the adjacent side to the angle, since by convention we measure angles CCW from the positive x-axis. The y-coordinate will be the opposite side to that angle.

[@gangan-sp9rf]

cos(5pi/3) Shouldn't it be negative 1/2 ?

[@potatomudkip]

Eyy my first comprehension that i did without any help

[@indiabiologyexplains5013]

If sin(7pi/2) is equal to sin(pi/2), which has a y axis of 1, why isn't the answer 1?

[@ZonguldaklıEren]

Nope it’s -1, replace pi with 180 degrees and 7.180/2 which simplifies to 630 degrees. 720 is the starting point minus 90. So you get to the coordinates of the bottom part which sin is -1

[@SaqibAlikhantanoli]

I don't understand what Cos, Sin and Tan(Theta) represent in first place? Is it angle, side length of triangle or what? I have been watching many videos to get the answer. Let me put it in this way: Theta represent the angle between two sides, then what does Sin(theta) represent? it is basically ratio of the opposite side to hypotenuse, but information it gives? same goes for other functions. Thanks!

[@ProfessorDaveExplains]

Start from the beginning of my trigonometry playlist and all will be explained!

[@Hoekstes]

I don't understand why the point at 1:56 is half pi. How does the circumference of the circle relate to the x,y coordinate? Edit: Oh, radians.

[@ProfessorDaveExplains]

Go a bit earlier in the trigonometry playlist, it's important to review what a radian is and how its derived.

[@siavashdanai7814]

Professor save me at 7:40, will you? Where did we talk about tan(x) being sin(x) over cos(x)? Didn't we talk of sohcahtoa only?

[@ProfessorDaveExplains]

Go back earlier in the trigonometry playlist where I introduce the trig functions and define them. But you can get it from SOHCATOA as well, (O/H)/(A/H) = O/A

[@topiado2073]

Sir plizz kindly share me the full course on the topic vector space☺ actually I would like to learn it from u only sir 🙏

[@ProfessorDaveExplains]

just go to my mathematics playlist! or even just my linear algebra playlist

[@topiado2073]

Okey sir☺

[@hemal6437]

Can anyone tell me , How the method applied here at 8.00?

[@emilyarmstrong5184]

But you are assuming that they know the angle measures in radians. Do you have a video that teaches that too?

[@ProfessorDaveExplains]

I do indeed! Check my mathematics playlist for everything I cover. Or there's also a shorter trigonometry series if that's all you need.

[@SaqibAlikhantanoli]

One more question: What if the circle is not unit circle? How would if effect the trigonometric function and their calculation?

[@gordonglenn2089]

Then you just need to divide by the radius to get the cosine and sine values.

[@carultch]

If the circle is not a unit circle, then you are adding extra steps. The circle is set up to be a unit circle, to keep the calculation simple. Given a circle of radius R, and an angular position CCW from the positive x-axis, the x and y values are determined by: x = R*cos(theta) y = R*sin(theta) Arc length from origin of angle = R* theta, when theta is in radians If you know x and y, and want to find the sine and cosine of the angle, then you'd have to divide by the radius of the circle to get the sine and cosine. This is why we use the unit circle, so that R = 1, and becomes a trivial term in these equations that we can ignore.

[@Benevolenttt]

🥰🥰🥰🥰 helps alottt💃💃

[@ZallyAimro]

such delicious pie

[@zlynnanneasuncion279]

Hello Professor Dave! Why is the bottom left problem's answer is -1? Isn't it supposed to be 1?

[@carultch]

The left side of the x-axis is considered negative, and the bottom side of the y-axis is considered negative.

[@toadamine]

The only provlem I have with this, is it's wrong wien I draw a circle and use a protractor to plot the angles and then measure the values with a tape measure, they don't match the formulas results.... It's a whole lot simpler just to figure out what the angle is and figure it's orientated of 360 then convert that to a measurement... this is just plain wrong wjen checked against measuring tools... The value for the first pint from zero is 0.5233... the formula is 2rPi/12... the formula for point two is 2rPi/8the formula for point 3 is 2rPi/6... then 2rPi/4... see how this works... also works using the percentage/decimal of the circumference.

[@fixtheminorissuesalso1217]

Unfortunately you're underrated i feel sorry for you😥

[@ProfessorDaveExplains]

please tell your friends to subscribe!

[@sainteilish778]

why not 1 but -1 sin7pi/2

[@derejehabtamu4427]

Best

[@mahmoudsalavati7939]

in the comprehension the answer of 5/3Pi is 1/2 but 5/3Pi is 300 degrees which is 270+30 so the cos should be root3/2. am I not right?

[@ProfessorDaveExplains]

no it's 1/2, check the unit circle

[@mahmoudsalavati7939]

@@ProfessorDaveExplains thank u for answering so quickly. :)

[@oy6648]

I’m a bit confused at 7:36 could someone break it down for me like what does 12pi/3 have to do with tan(14pi/3) and why did it randomly appear ? I might be a bit too dumb to understand. Like I’m talking baby steps

[@YashGoud-d6k]

I understand it too I got it after brain storming a bit so here it is this is my 2nd time writting , this the shitty yt algorithm can't even save the msg Here I begin The value of 14pie/3 is not in it's simplest form so we subtract 12pie/3 or 4pie you will get 2.09523809523809523809523 this You should try it out too in calculator get it out give it a try and next to confirm your answer Type 2pie/3 and you will still get 2.09523809523809523809523 this

[@oy6648]

@@YashGoud-d6k thank you 🙏

[@topiado2073]

-17π/4=16π/4 how sir?

[@adhiyanthaprabhujeyashanka2091]

Bro he is not telling that -17pi/4 equals 16pi/4, he is telling that 16pi/4 is 4pi, and that is twice over the unit circle, and when you convert -17pi over 4 to degrees, you will get -765degrees which is the same as -45 (co terminal angles)

[@sahasavadmangkhaseum5455]

I don't under stand from 7:24 onward; How you do it? Why do you need to mention 12pi/3 and subtract it to 14pi/3? WHy? Thx pro dave

[@ProfessorDaveExplains]

12/3 pi is the same as 4 pi, or two times around, so 14/3 pi is the same as 2/3 pi, etc.

[@sahasavadmangkhaseum5455]

@@ProfessorDaveExplains I still don't get it Can you go even deeper Thx Appreciate it!

[@sahasavadmangkhaseum5455]

@@ProfessorDaveExplains I finally got it? 1 more question. Isn't 7/pi2 equals to 1 not -1

[@ProfessorDaveExplains]

how do you mean?

[@sahasavadmangkhaseum5455]

@@ProfessorDaveExplains Oh i see what you did there sorry for asking a stupid question. Anyway, I appreciate a lot Dave sensei

[@codexcodexcodex]

Is that the way calculators calculate sin x and cos x?

[@learncraft8486]

👌

[@Benevolenttt]

Woooo

[@mrbottler9627]

Thank you maths Jesus

[@potatomudkip]

why π and not tau

[@carultch]

Historical reasons. Pi was the constant that has been defined for the circle for thousands of years. The idea of using tau to replace pi, is relatively recent, and originated with proponents of reform to our conventions in mathematics. You will only confuse people if you insist on using tau, without specifying what it means, since pi is the letter with a conventionally understood meaning. If you prefer tau, you can write "let tau = 2*pi" at the top of your work, so that others know what you mean when you write the letter tau. Then translate your answers in terms of pi, so the person reading your work can see your answers in a form they understand.

[@HtetKaungSan-eo6sq]

something is not right

[@ProfessorDaveExplains]

no it's all good

[@eon1311]

Idk why I’m here even tho I’m 11

[@ApdiHakiim-gt1ug]

My english is bad so is liitle hard understanding u😐

[@HassanDinbalidinbalihassan]

ابله می داتی عدد پی غلط است

[@stevenwilson5556]

"Just stare at this until it makes sense…" sorry but this is garbage advice. You did a decent job of explaining this but if people don't understand from first viewing, staring at it won't help. Maybe watching the video again might, but staring won't.

[@ProfessorDaveExplains]

Examining something and analyzing it until you understand it is a trivial and common sense approach to comprehension.

[@Chaotic369]

Everyone got their brain and no one is blindly following anyone's advice if one is unable to understand something again and again then shall analyze things with a different perspective and look at something and thinking about it until one get it that what I think and this works for me takes a little time but does

[@SuperMan-uj7dh]

Hey there!! I'm first!!

[@chayanmajumdar8195]

Haaha

[@belgiumball2308]

Lol