All the TRIG you need for calculus actually explained

[@DrTrefor]

What trig formula do you find the most useful???

[@studentsofmathematics8995]

Sir I am MSc mathematics and I wanted to work with you. From Pakistan

[@gregwochlik9233]

All of them have their uses. A carpenter does not have a favourite hammer.... wait, that is not right.... I say (sin x)^2 + (cos x)^2 = 1. I have a collection of 2 slide-rulers. on the bigger one, the pythegoran idenitity features.

[@josephshaff5194]

I'd like to bring a Mathematical Shape into CAD that shouldn't be broken up into lines and it must be true to fractions of wavelengths of light in profile. Not be broken up into small Line segments in profile. It's like a Parabola. Any recommendations ? It's not for work. Just an experiment.

[@emanuellandeholm5657]

Euler's

[@MrMiguelChaves]

e ^ iφ = cos φ + i sin φ

[@Darkev77]

Ooooof, I really forgot most of my HS trig stuff as I went to undergrad and grad school. I was like it’s about time I refresh my trig knowledge, and now I get Dr. Trefor’s review. Insane!

[@DrTrefor]

It's not so bad to refresh imo:D

[@Darkev77]

@@DrTrefornot bad at all, especially coming from your explanation! Thanks a ton!

[@Fracasse-0x13]

You're in grad school and didn't know this?

[@OcaRebecca]

I’m a minute and a half in. You already did more to relate the unit circle to triangles than my precalculus teacher did some 25 years ago. All the unit circle was was merely a bunch of points to be memorized. Trig is a special form of hell when your teacher is constantly absent or tardy.

[@shaileshsankaran3021]

True that...the love for maths is directly proportional to the kind of explanation that you received from your teacher in your school days...if everyone were taught trig like this ...we would have far more STEM grads

[@Lhosal]

I was a pure math major, didn't finish yet but I just need about 3 classes, I noticed that a lot of students in the math degrees (pure, applied, and education) struggled with a lot of fundamentals that they learned but didn't really understand in their lower education. So I didn't see students struggle with only trig, but also a lot of algebra (partial fraction decomposition, factoring, manipulating equations), exponents and logs etc. They're all essential in the most failed course during university, the class that brings it all together... calculus 2. Because it's so essential to be able to look at a cal 2 integral, and see within it, a way to manipulate it into a integrable form.

[@acrane3496]

It’s also because we don’t get enough time to master these fundamentals. It’s one topic after another and you don’t even know why they’re important at the time of learning but after doing advanced math, it’s easier to understand and remember. Like i sucked at linear algebra the first time and found it boring but after taking abstract algebra, retaking linear algebra is so easy peasy but it’s still slightly boring though. But math is so easy when your foundations are strong and imo mastered. But my foundations are weak so i struggle. And math is extremely foundational. No other subject matter requires this much foundational knowledge and mastery.

[@Lhosal]

@acrane3496 I agree, it's also about the types of questions that you're essentially forced to answer, they're very formulaic and it continues that way until Cal 2. Even Cal 1 is just, apply the ***** rule, like the chain rule or the product rule etc with very specific rules. Then... you hit Cal 2, and all of a sudden, you need to look at a fractional polynomial, decompose it, factor, apply very old rules to very difficult new questions and then a bunch of new stuff, and you're expected at that point to start thinking of math as an language and a solution as a statement or and entire narration, where you lead the reader into understanding why you applied what rules, why it works, and then get, sometimes obscure answers. If at any point you have a misunderstanding you compound that mistake. I think I had an advantage in university because I started late, a full 10 years after graduating from high school. And I didn't have any math related jobs, just an interest in physics. I didn't want to have to retake classes I took like trig, precal, college algebra etc, so I self studied, but like seriously, with khan academy, with books, solving a lot of problems, and it was a lot of stuff I already knew, but a lot of stuff I didn't. I think maybe it's because I started in physics and moved over to pure math, but... there's really something behind the idea of just practice, math takes a lot of practice. When I was a physics student, the culture in the program was just "solve a lot of problems" and it works, it made me really good at basic arithmetic which heavily carried over.

[@NoDrizzy630]

I’m finally going to college at age 45 and forgot all these things so I decided to take remedial courses(my employer is paying) but I want to truly understand these subjects not just struggle through it. I believe having a solid foundation will make harder maths much easier to digest.

[@user-ug2vw9vb2v]

i'm so happy to see that this guy is finally getting the sponsorship for all the hard work he's been doing. Thank you professor!

[@TommasoGianiorio]

Rationalising can be useful for a fun memorization of the values for the standard angles: Root(0)/2, Root(1)/2, Root(2)/2, Root (3)/2, Root(4)/2.

[@epsilia3611]

6:42 One reason I found is how you have a nice pattern by doing that over the particular values : sqrt(0)/2, sqrt(1)/2, sqrt(2)/2, sqrt(3)/2 and sqrt(4)/2 are the five main values we see in high school when related to trigonometric function sin and cos, so it makes up for a very good mnemotechnical thing, so why not it's always good to take

[@chaotic-ilusium8363]

I like the "sqrt(2)/2" notation more because it's nicer to remeber : when you are closest to the center it's 1/2 or sqrt(1)/2 then in the middle it's sqrt(2)/2 and finally the closest to the edge is sqrt(3)/2 this 1 2 3 patern helped me understand and not have to know the vallues by heart

[@gregwochlik9233]

Nice. I high school (South Africa 1992 to 1996) I had just the few identities memorised. The rest I derived. For example, I memorised cot = cos / sin, as the two "c" went together. I remember how I "discovered" the sin(60 deg): I took my calculator in a plastic bag into the bath. I asked for sine 60 degrees, then I randomly squared the answer. I was astonished to have gotten 0.75. So, my sin(60) = sqrt(3/4).

[@GregSpradlin]

That unit circle diagram you showed at the beginning is the one I show my students. I call it The World's Greatest Unit Circle Diagram.

[@steve6012]

if you remember Eulers and Pythagorean and that Cosine is even and Sin is odd, you can sit down with a blank piece of paper and easily (no geometry) derive angle sum, angle diff, double angle, half angle from scratch

[@ZaksLab]

haha! I struggle to convince my calculus students to say 1/sqrt(2) instead of sqrt(2)/2 on a regular basis. I had a colleague explain to me early in my career that the motivation for rationalizing everything is that numerically approximating these expressions with paper and pencil is way easier with a whole number denominator; i.e., math teachers are insisting on rationalizing these expressions because 50 years ago it was considered important to be able to convert radical expressions to decimals by hand! I currently tell my students the history of the thing, but I also point out that being able to manipulate radical expressions is still a valuable skill . . . when you actually need it.

[@ColinPaddock]

It’s fairly easy to see that 2x(sqrt(2)/2) equals sqrt(2). It’s a bit harder to see what 2/sqrt(2) means.

[@ColinPaddock]

Getting used to the manipulation can make it easier to see relationships like that.

[@johnnolen8338]

The angle addition formulas are derivable almost instantly from Euler's formula: e^(iθ) = cos θ + i sin θ.

[@DrTrefor]

Ya this is a great “starting” spot, although a little higher level than my target audience here coming from high school

[@ronkirk5099]

Because It had been a while since I took Trig, I took a refresher class concurrent with my first semester of Calculus. No much Trig required for Calculus. Great math T-shirt by the way.

[@bigprogramming579]

Oh god finally a video which explains all these topics so easily, tysm!

[@DrR0BERT]

Dr. Bazett, when I teach the trig of standard angles, I tell my students to rationalize their denominators for the following reasons: * It is an easy way to remember the trig of the standard angles √0/2, √1/2, √2/2, √3/2, and √4/2 for sin(0), sin(π/6), sin(π/4), sin(π/3), and sin(π/2) respectively. * When working with the radical fractions, it is easier to work GCD's when the denominator is an integer over a radical. For example, √3/2 - 1/√2 is not obvious how to combine. * When computing tangent and cotangent of the standard angles, I tell my students to ignore the 2's in the fractions. I remember when I was learning trig back in 1980, before scientific calculators found their way into our high school classrooms. Our high school math teacher had us divide 1 by 1.414... using long division. She then rationalized the denominator and showed us dividing √2 by 2 was a lot easier. Now with calculators that's no longer an issue.

[@chrisbrown865]

thankyou good introduction to trig identities

[@DrTrefor]

Glad it was helpful!

[@Puguinsquatch]

I was just studying until I noticed his shirt I LOVE IT-

[@wolemai]

It took me a couple of seconds to realise what your t shirt was about. I love it! Of course the hypotenuse is the hippopotumus. Why have I never heard that before? That cute hippo will help make trig more enjoyable.

[@DrTrefor]

Haha isn’t it fun:D can also do “hypoteMOOSE” and put a moose

[@AlberTesla1024]

This is great, I am getting the perspective of the values of right angled triangles especially how the values are given for specific radians, which I memorized in school.

[@BrianDominy]

When I was in high school I had CSH written in the front of my notebook, shorthand for "cos 60 = 1/2". From this I knew if I changed the cos to sin, or the 60 to 30 degrees, it would be the other value of sqrt(3)/2. I found this the easiest of those 30-60 angles to memorize.

[@DrTrefor]

Cool mnemonic!

[@wintutorials2282]

The timing is incredible

[@user-zy6ko7zf5l]

Thank you, this was very well explained.

[@harrysolas2802]

Thanks for reframing this information. I suspect that watching the unit circle spin is related to a spinning magnet in electrical engineering. All the math you need for Maxwell's equations would be a great series.

[@JimRohn-u8c]

Thank you so much for this!

[@AzureSpike]

rationalizing the denominator is needed for easier division, like dawg i am NOT gonna divide 100000 by 141421, id rather divide 141421 by 2

[@DrTrefor]

I mean sure, I liked that reason before calculators were around, but unless I'm trying to approximate the exact value by hand I'm not bothering:D

[@dono42]

The origin of rationalizing denominators is when people used to do division by hand before calculators existed or were became common. It is much easier to calculate a rationalized fraction than a non-rationalized one. The problem now is that teachers have since forgotten why this started and hence still teach it when it has very little value in our modern society anymore.

[@andrewharrison8436]

I don't need this, but am I delighted to have it - yes. So long since I saw the derivation but I am fairly sure it wasn't as elegant a piece of geometry.

[@kmunson007]

Fantastic!! Thanks for sharing this. Just right in all respects.

[@columc]

Most helpful math video ive seen

[@PlasmaCoolantLeak]

Closet nerd here. Seriously digging the tee.

[@maxsilver4197]

Rationalization of denominators is useful for both division via slide rules and doing division by hand, in the case of sin(pi/4), working out 1/1.4142... is a lot more work by hand than 1.4142.../2.

[@maxsilver4197]

With the widespread adoption of handheld calculators, I do agree that rationalization of a denominator is largely unnecessary in most situations.

[@DrTrefor]

Ya I think this is exactly why it used to be be completely standard.

[@vimaljain7950]

Nice explanation❤

[@BramCohen]

In high school I had a math teacher who turned all math into dreary awful. She had a whole section on proving trig identities. On every test I'd write out e^(it)=cos(t)+i*sin(t) then derive the complex formulas for cos and sin then plug those into every identity and simplify them all. And she marked me wrong on every one.

[@DrTrefor]

Amazing

[@geertdejonge4194]

With Euler's identity, complex vectors, matrix multiplication en there and back again trig is much more fun.😛😛

[@SuperDeadparrot]

I think rationalizing the denominator comes from the days of slide rules and log tables. It’s easier to divide root 2 by 2 than to divide by root 2. Now that we have calculators, that issue is gone.

[@zbaltazard]

All the trig info you need for Calculus *Sees 20 min video* "Yea that makes since" - Calc 3 survivor

[@katiyas8353]

love the shirt!!

[@stewartbrown7907]

A note about why we rationalize denominators (or so I'm told) goes back to when there wasn't electronic calculators and you needed to look up values in tables. These tables would include sqrt(2), but not 1/sqrt(2), so making all the roots show up in the numerator allowed one to continue doing their computations. Do I think we should still teach it, not really.

[@eceto123]

Great vid!

[@zachsilveira4580]

Great video

[@josephshaff5194]

Thanks for the video.

[@DrTrefor]

You're most welcome!

[@jackkalver4644]

I have a proof that d/dx sin x=cos x that basically derives the usefulness of radian angles.

[@sdspivey]

When in class, do not memorize. Learn!

[@DrTrefor]

This is just a the best lesson for basically everything in math

[@marekglowacki2607]

I think "rationale" ;) for rationalization 1/sqrt(2), could be this: sqrt(2) / 2 = 1.41... / 2 = 0.7... easier to calculate than 1/sqrt(2) = 1 / 1.41...

[@cparks1000000]

"Pre-calculus" needs to be abolished as a class and replaced with algebra-for-calculus and trigonometry-for-calculus, both full-semester classes. Students who are comfortable with algebra, can move directly into trigonometry. Students who are comfortable with trigonometry can just take algebra.

[@culater]

Dear Doc Trefor, is the double angle formula shown in your video correct for cos? If so, can you please explain it to me ? (At about 14:55 in this video) THANKS !

[@prycenewberg3976]

You've lost me by 0:46. My biggest question is, "Who invented the Unit Circle and what problem was he trying to solve?" The Unit Circle seems like i (a convenient myth we can use to reach answers) but no teacher has ever explained it like that, and I don't know if that's right.

[@DrTrefor]

The original goal was likely to do with studying phenomena in nature that have a periodicity to them (like say the height of a dot on the edge of a wheel)

[@information5245]

We are waiting for you to prepare content on the topic of differential and integral equations

[@DrTrefor]

I plan to do more! I have a playlist on ODEs that this semester will definitely be expanded.

[@ProactiveYellow]

Speaking as a math tutor, I find rationalization of a denominator to be "nicer" for my students to think about. When we say 1/√2, my students interpret division as "cutting into parts" so how do we cut it into "square root of two pieces?" We can't always make the denominator a natural number, but doing so makes it so that we're taking "half of √2." Another example is that we could totally talk about 5/(-i√3), but there's so much "happening" in the denominator that it's easier to clear it out a bit and rewrite it as 5i√3/3 because numerators tend to be more "flexible" than denominators (the negative disappearing because of i² in the denom when rationalizing). Is it mathematically necessary? Not really. It is helpful for making the most "algebraically accessible" representation of the number. We have to remember that math is performed by people, and notation should reflect the intuition of people to reduce confusion when possible. You and i are comfortable putting any number anywhere (except perhaps 0 in a denominator!), but this only comes with years of practice.

[@DrTrefor]

I think something like this was part of the historical preferences, that root 2 would be a memorized number and then it is easy to take half of a memorized number.

[@ProactiveYellow]

@@DrTrefor absolutely. Another historical preference has been pervasive here in using "√2" instead of 2^{1/2}. We could write out everything as products with exponents and use negative powers for division, but historically roots and exponents were perceived as different, and that perception has led to how we interpret expressions today. I went to college for music, and there are many expressions there which technically mean the same thing but are written fundamentally differently just for making it easy to read as you go. I see the same preference for legibility here, and the traditions that inform what is legible are quite interesting to observe.

[@motasam8122]

greatest

[@irvyn1430]

Good video 👏

[@diribigal]

A nitpick: It's a bit of a pet peeve to see words like "Circumference" and phrases like "Arc length" written in italics with no spaces, as if they were products of variables. Of course, no one is going to confuse the word "Circumference" with the quantity "c^2 C e^3 f i m n r^2 u", but upright text would prime me to read it as a word more easily. That aside, this is still a great video and reference for calculus students!

[@DrTrefor]

I always check over my slides to see if there is something small someone will be annoyed by, I can't say I would have ever thought of this one:D

[@Don.Franco_Film]

0:54 looking a little like a spiral there!

[@Logic_Lessions]

Sir , Which software did you use to create such animations?

[@donaldspanyer6403]

Minor note: At 2:21, "circumference" is spelled incorrectly. Otherwise, nice video! :)

[@ikocheratcr]

I always swap csc with sec :( In my mind csc starts with "c", so it should be the 1/cos() one, and sec starts with "s" so it should be the sin one, but no, it is the other way around, sucks. I know it is a naming convention, but makes me make mistakes.

[@DrTrefor]

it really does seem counterintuitive just based off the names. I do like that the triangle that defines sec also defines tan, and so then csc and cot pair off too.

[@ucchuman]

please answer why everything becomes "e" raised to something in college? why dont they keep curves presented with sin and cos

[@deniseockey6204]

THIS IS WHAT IS WRONG WITH OUR MATH AND SCIENCE COMPETENCY OF STUDENTS IN THIS COUNTRY. THE CURRICULUM WAS BEEN SO DILUTED THAT STUDENTS TODAY ARE BEHIND THE REST OF THE WORLD. TRIGONOMETRY USED TO BE A FULL YEAR. NOW AT MOST IT IS A CHAPTER. THOSE WHO ARGUE NOW STUDENTS ARE TAKING PRECALCULUS IN HIGH SCHOOL. I TUTOR THESE STUDENTS. THESE CLASSES ARE JUST WATERED DOWN ALGEBRA TWO. THIS ISSUE GOES FAR BEYOND MATH AND SCIENCE. CURRICULUMS ON MANY SUBJECTS ARE WATERED DOWN OR KEY COURSES ARE ELIMINATED ALL TOGETHER AS WE EXPECT LESS NOT MORE. PATHETIC!!!

[@tiusic]

If I could delete one thing from trig education, it'd be the secant/cosecant/cotangent stuff. It's just more confusing rote learning, and I've never used them in real math. In fact I don't think it even came up on the exam for that class 😂

[@gilbertley1309]

Secant is really nice for trig subs when integrating because 1+tan^2(x) = sec^2(x)

[@user-zz6fk8bc8u]

And now let's switch 2π for τ and it will be even simpler 😅

[@DrTrefor]

haha I do like a good tau

[@VideoFusco]

What exactly do you mean by "everything you need"? What you have presented is certainly a complete compendium of all the necessary definitions, properties and trigonometric formulas, but it seems to me that in calculus it is also necessary to know how to solve equations and inequalities of various types. And although, in theory, all the techniques for solving trigonometric equations and inequalities come from the formulas and properties you have shown, the average student is certainly not able to think up or reconstruct such solution methods on his own unless someone has first explained them to him and has had him practice properly on certain things.

[@DrTrefor]

I mean something like the basic trig facts one (commonly) uses in calculus. That isn't to say there isn't plenty of other precalculus skills like how to solve equations and the like, logarithms, graphs, inequalities, etc etc. I might do some videos on other precalculus concepts, but this one is just meant as a refresher on sort of trig in isolation.

[@reidflemingworldstoughestm1394]

Trig is the easiest math class you will ever take.

[@GrifGrey]

kindergarten math

[@dean532]

I like your T-shirt

[@OpPhilo03]

Sir, why we take Unit circle?! Why not we take in general any length of radious circle?!

[@DrTrefor]

We could have defined these functions with any radius, but it is cleaner to do it with a unit radius because whenever you have the hypotenuse in the formula (like opposite/hypotenuse) you don't have to worry about those denominator. If you did, say, radius 2 then there would just be a stretching factor of 2 everywhere.

[@AlberTesla1024]

Those come in handy when we work with vectors especially when we just need direction only which is basically vectors of unit length called as unit vectors.

[@algirdasltu1389]

Because if you know these on tge unit circle, you know it on every circle, all you gotta do is scale by the radius.

[@Nomedecor]

13:30 that one i didn't understand

[@rupom_1670]

hey Dr. could you please explain or provide me with a video why calculus is 100% precise? i mean i don't quite get it why it would be 100% precise if we say the approximation of something in calculus gets more and more precise as dx tends to 0 anyways thanks for the great video Cheers 🥂

[@MrVontar]

I only stick with 1+1=2, hehe. 1-(-2)=3.3/1.1, ez mode.

[@26IME]

What *trigs* me more is Wikipedia on clear/bright mode

[@DrTrefor]

lol harsh but fair

[@eldersprig]

weird trig functions: chord, versin, haversin

[@wobaguk]

Was taught Silly Old Harry Caught A Herring Trauling Off Afghanistan, when I was about 13, was that unique to the UK?

[@scophylio1]

It would be better if you had a board and worked it out for people who do not understand.

[@Dominus-F6uZr]

I want the shirt so bad :)

[@AhFeiLee]

I want a t shirt like that...

[@kdi9049]

You lost me after the first triangle

[@jjspr]

No tau fans here.

[@samin21]

Method 1) (- x= 3) equation is given Multiplying both sides by (-1) -1*-x=-1*3 Then x=-3 or Method 2) Let the equation be (- x= 3) If we multiply both sides with "MINUS" sign -(- x)= -(3) Then x= -3. Which one is correct or both methods are correct . Please help 🙏🙏

[@vimaljain7950]

Mathematically method 1 is correct but method 2 is also Same as method 1 here, Multiplying both sides by minus sign it means " minus One"😊

[@maxsilver4197]

Both are equivalent in most cases

[@michaelrafa100]

Slow your talking down...

[@kadensmith5586]

Nahhh, Ima put this in 2x - maybe just put it at 0.75x, but lowkey it does seem like he sped it up, but because it was probably like a 40 min vid before he did

[@klarion]

Why... The first question is always why and what for... But 99.9 % of math teachers fail at that. To me, without why, math is meaningless... A bunch of pointless, pun intended, characters.

[@908animates]

Please dear God someone tell me how it works not how to memorize a bunch of fucking graphs

[@antonipeters2333]

Fruggin'ometry!