Using the Magic Hexagon to Generate Trig Identities
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- Опубликовано: 30 июл 2024
- This video demonstrates how you can use the Magic Hexagon to generate commonly used trig identities. This version of the Magic Hexagon, as well as many other versions can easily be found around RUclips and around the web.
#mathematics #trig #precalculus
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Omg you saved my life from getting a bad score on a pre calc quiz I swear it was the last min and I hadn't studied for this till I have discovered your video please tell me some other tricks on how to do other topics GOD BLESS YOU I WAS ABOUT TO CRY
I have a playlist for various tricks and short-cuts. Here's the link: bit.ly/2WV1QFd
i wish i could have found something like this as a kid. thanks for this mam.
Thanks for watching!
Studying for the FE Exam after 7 years out of college, and this just blew my mind. Thank you!
thank you so much!! I was told to memorize this with no way how.
you were a gift from god!!
thanks jesus
Thanks for watching!
Youse Jose is c
👍🏻♥️😀
Excellent. I wish i had a friend to teach math to me like this. This is so easy. Thank you so much.
So glad the method worked great for you! Thanks for watching.
We need more people like you! Thanks and thumb up!
+Horror Time Thanks! Glad it was helpful!
My school never taught me this
It’s definitely not in any curriculum. It’s just a way to memorize trig identities. Thanks for watching!
I dont know how to thank you about making this video. This is the yewel in the set of all math-memos, and the most useful one also. Maybe I just put a hearth symbol here to show my gratitude❤ Thank you!
Glad it was helpful!
wish my pre calc teacher taught me this... this is such a big help thank you!!!
Glad the video was a big help to you! Good Luck in your pre-calc class.
Wow, beautiful symmetry! Thanks for sharing!
thanks a ton! this really helps out when u don't have the slides from the presentation😊
I’m studying for the precalculus clep test and I had forgotten just about everything from trig when I took it about 10 years ago. I’m so happy I found this, and when I eventually become a math teacher I will definitely refer my students to this video!
Thanks! Good luck with your upcoming test.
This is sooo useful, thanks for the video, also good explanation!
Glad you think so!
Thank you so much:) I have cal 2 trig integrals and this is perfect since you no longer have to memorize all the formulas by heart:)
You're welcome. Thanks for watching!
Excellent explanation! Tremendous help.
Thanks for the great insight. You are a gift to humanity.
Thank you so much!
Thank you so much, I'm a first year engineering student writing Calculus tomorrow and I know this will help me a lot! :)
Glad it was helpful! Thanks for watching!
Thik video helps me very much to score good marks in trignometry .If you all are agree then like this comment.
Glad the video helped you! Thanks for watching and commenting!
This was very helpful, thank you!
You're Welcome! Glad it was helpful!
Fascinating! Thank you so much!
You're welcome! Thanks for watching.
This is cool! I'm not sure it would have worked well for me, but I can see some students for whom it would be a thing.
Here are things I know, and I build the rest off of it:
1) Oscar had a heap of apples, or SOH CAH TOA, or Some Officers Have Curly Auburn Hair To Offer Attraction.
2) It's a right triangle with a hypotenuse of 1, and the Pythagorean Theorem applies, so sin^2 + cos^2 = 1.
3) The csc / sec / cot are just reciprocals of sin / cos / tan. The reason that "cosine" and "cosecant" aren't reciprocals of one another is because somebody hates children and made it needlessly difficult.
4) Complementary angles.
If I memorized the hexagon, I could probably get to usable results quicker. But I also feel like, this way, I have a solid understanding of how to get from one point to another.
very nice trick ..... made my day!!!! lots of blessings and subscribers to you>>>>>
Thank you so much!
Wooooww thank you sooo much for this!
You're welcome! Hope it was very helpful for you!
I think 5 people who disliked this are from another planet. Btw this is amazing . and the best maths video i ever watched. Thank you very much........
Thanks! And thank you for watching!
Thank you so much ....helped me a looooooot....deserve a subscriber
Thanks! I have lots of playlists on my channel that cover a lot of different math topics. Be sure to check them out.
thankyou very much . I really appreciate this
You're welcome! Glad it was helpful. Thanks for watching and please be sure to share with friends.
Do you also teach the Magic Hexagon with Differential and Integral Calculus for differentiating and integrating along with Hyperbolics? Thanks for the video.
Thanks for watching. I don't use it for Hyperbolics, but only because I don't teach anything higher than Calculus 1.
Excellent!
All the other videos on youtube but none of that beats this
Thanks! That's great! Thanks for watching.
Great video, very helpful Thank you.
You're welcome!
wow now this solve my challenge of trying to memorise everything.... thank you
It is definitely a handy method to use! Thanks for watching! Please share with your friends.
Jeeeez!! Thanks much for all these!!!
You're welcome! Thanks for watching.
thankfull to u, have made me understand
Thanks for watching! 😀
Thanks for making the work easy for me.
You're welcome! Thanks for watching.
hey,
this one is very good for my project on trigonometry..
but i want to also know about the co functions explaining supplementary angles..
please help me
The only video I have that deals with Cofunction Identities is a short, straightforward video that just shows how to find a cofunction angle. Here's the link to that video: bit.ly/2TbG9i6
Very very useful thanks a lot..... keep doing....
Thanks for watching!!!
Thank you so much, does these kind of stuff common in USA? They never taught us tricks like that in Turkey.
This kind of stuff is not common in any school. Ülkene özel bir şey değil, okul işi değil bunlar.
@@keilast7503 Thanks for your answer.
It really just depends on the teacher. Lots of teachers in the USA give mnemonic devices for memorizing formulas. And that is all this is.....just a way to help students memorize Trig Identities.
This is gold!!!
Great! I'm glad it's helpful! Thanks for watching.
Fantastic! Brilliantly simple and so useful. Great work!
😱😱O god where were u this whole time 😳
Right here! Be sure to check out the rest of my channel. Everything is organized in Playlists.
Thank you. thank you
is it possible tanx=sinx\cosx is desame with tan=sec\csc? since you told us counter and clock wise are appropriate.
Yes......sec/csc will ultimately simplify to sin/cos which is tan.
I wouldn't even waste resources remembering that I have to go counter clockwise when I use the inverse function(tan -> cot), I mean it's the inverse function. You just swap tan and you get cot.
Very True.
Yes, those are the easy ones to memorize, as is sin²+cos ²=1.
Tan and Cot go together. The others swap both the prefix and the rest of the word, so sin goes with a co- and isn't cosine, therefore it's csc. Cosine woul lose the co and can't go with sine, so it's secant.
It's like tips on fingers, so kind of you Madam.
Thank you! Thanks for watching!
Experts and those who have competed calculus, is it better to memorize this or derive them a billion times to memorize that? Knowing these let's you go one direction, but it doesn't seem as easy to see a problem and know the next step unless you memorize the formulae instead.
I love this so much thank you
You're welcome! Thanks for watching.
Nice Gerome
HAHAHAHAHA nice
Awesome!!!
Thanks!
This video is just the best
Thanks! Thanks for watching.
When was this "magic hexagon" invented? I don't see it mentioned in Wikipedia articles for trigonometry.
Honestly, I'm not sure when it was invented. I know my high school & college years spanned 1979 to 1987, and I was not introduced to this back then. I first saw it on the internet a few years ago, and decided to use it in teaching my classes.
It was published in 1978 in Chinese publication named " Mathematics Handbook".
This is so freaking good
Thanks!
Cool video! 😊
Thanks! 😊
My god I would love to have known this when I was in undergrad. Mind blown!
Mathematics is patterns!
thank you so much you saved my 15 marks thank a lot
Happy to help
thanx for ur reply
Thx so much
You are welcome!!
Good work
Thank you! Cheers!
So, in the quotient identities, tangent can also equal sec/csc ?
That should be correct.
If tan = sec/csc,
Tan = sec/1 • 1/csc
Since /1 equals the numerator, and csc (reciprocal of sin) is under 1, it reciprocates again.
Tan = sec • sin, and we see that matches one of the multiplication rules.
We then can also see tan/sec = sin, tan/sin = sec, and (sec•sin)/tan = 1
You can keep manipulating and swapping terms all day around the whole triangle. What this hexagon does is show that same triangle in 6different perspectives.
Very interesting Thanks
You're welcome!
This deserves a lot more than 58k views.
Thank you! Several students really do like this method.
Thanks!!!
You're welcome! Thanks for watching.
how do you solve it ?
Very nice
Thanks! Thanks for watching.
Brilliant
Now I can use it for my maths H.W
Great! Thanks for watching & commenting!
Thanks
You're Welcome!
I like this videossss
Thanks for watching.
I'm John Daker, and that, my friends, is Amore.
Thanks!
good job
Thanks!
+Bharat Basel Thanks!
Holy crap I love you
Thank you! Glad you enjoyed the video!
lookitcrashed goot
How about derivatives?
wow. thanks
You're welcome! Thanks for watching.
This is really good. Why is it so hard to memorize trig?
Thanks! I agree, students do seem to struggle with memorizing trig functions......it’s definitely a mystery!
Cole's World of Mathematics Most teachers dont show the pattern behind it. These identities aren‘t even teached in my country...
Is this considered cheating if i write this down on my hand during a test?
Within my own classroom, I would consider that cheating. However, I would allow a student to have the Hexagon memorized, and then write it down on the test as soon as I gave the test to him.
ms hudson made me watch it
Hope you learned some good tricks for remembering the trig identities.
great
Thanks!
What is this sorcery?
Good stuff! 👍🏻👍🏻
reminds me of "horadric cube"
Now that you mention it, yes it does!
next slide on Pythagorean identities how about this:
you may never add from the bottom.you may subtract the bottom from the top to get a negative of the other top.you may add two negative tops to get a negative bottom.let me know what you think.
رائع
شكرًا لك
Jesus fucking christ thank you so much
You’re welcome! Thanks for watching!
Here in the philippines we called it piattos
Thanks for sharing! That’s interesting!
No hablo ingles mi amigo.
Same fam
Lo siento. no hablo español ¿Ayuda el subtitulado?
Marry me!
🙄
Awesome! Now show me where I'd use these anywhere outside of a math class...
+Go Get Some Trigonometric identities are found heavily in architecture, especially when developing large infrastructure. It is common to find them when constructing blueprints for actual structures. The reason that trigonometric identities are so important to architecture is that it helps you be as accurate as possible when determining sizes and proportions. This lets you know how much space you have to build, as well as what is the most efficient way of building something so that you maximize space, while minimizing cost, which means maximizing efficiency in construction.
Trigonometric identities are also applicable in the field of music for stringed instruments. For example, the vibration that a violin possesses is the same shape as a sine function. When playing instruments you don't think about trigonometric identities, but when calculating the physics behind it, they come into play. Trig identities in music are typically a calculation of frequency and are represented by using kilohertz (kHz).
The six trigonometric identities are very prominent in several different types of engineering. Some of the more popular types of engineering where trigonometric identities are used are civil, electrical, and mechanical engineering. They are used when analyzing alternating and direct currents. The shape of the curves that alternating and direct currents form correspond to that of the trigonometric identity curves.
In general, trigonometry itself is helpful for things like separating forces into orthogonal components. The concept of force is used in all sorts of kinematical applications. The trigonometric identities simply assist in the manipulation of the trig.
This is just a few applications of Trigonometry and Trigonometric Identities. But hopefully this gives you some idea how it can be used outside of the classroom. Unfortunately, you have to take several years of "classroom" mathematics in high school and undergraduate courses before you have an opportunity to apply what you've learned in the classroom to any actual real world applications.
Which is exactly why so many people fail at math and don't remember anything past test day. I've had a love/hate relationship with math my whole life. Mostly I hate the way it is taught with theory first, practical application MUCH later. We don't learn music this way. We grow up hearing it and singing (and humming) along way before we learn about notes and measures etc. Juggling has a notation, but no one learns that before picking up a ball. It is ridiculous that we do this with math. Most math "teachers" are not very good at getting this across in a MEANINGFUL way. We need to fix this if we want more people to love all of math.
I agree, the way math is taught does need a total overhaul. But there are teachers scattered throughout the US that are presenting mathematics differently. For example, Dan Meyers. You can do a search for his TedTalks on RUclips, or even Google his blog. He is inspiring other mathematics teachers around the globe to change the approach to the way mathematics is taught in the classroom. But that change won’t occur overnight, it’s going to take time for that change to occur. Currently, US teachers have to worry about standardized testing, which is important in the fact that those test scores determine the amount of funding schools get from the government. And schools can’t exist without the appropriate amount of funding. Unfortunately, it’s a vicious cycle that is drastically affecting the way students view education in general, not just in mathematics.
Currently, I attempt to combine traditional learning of mathematics (“crank out answers to problems”) along with applications of the mathematics (or at least the best way it can be simulated within a classroom environment). Unfortunately, the only thing on RUclips people see is the traditional “cranking out answers to math problems”. My guess is because the majority of the mathematics teachers are still teaching in that mode.
Obviously, the problem is not an easy one to solve, but I do understand your frustration with the way mathematics is taught. If at all possible, hang in there, because if you make it far enough into the study of mathematics, there is a point where it becomes truly beautiful!
After teaching undergraduate math for over 20 years I've stopped answering this question. I reject the premise that math needs to justify its existence with demonstrable applications. It's like asking how language is useful. For starters you couldn't ask the question without language. You couldn't even think of the question without language. No one asks a language teacher "when am I ever going to use verb conjugations outside of a language class?" At least if they did the answer would be pretty simple: walk out of the classroom and start talking.
Math is the language of science and engineering. It's the context in which these things are thought about and accomplished. If you want to work in international business in a foreign country, you must first learn the language used in the country. When you do the work of business in that country, you'll be using the language all day long, every day, to talk and read and write about lots of different things. And when you do that, you're focused on the meaning of the language in the particular context, not the language itself. It's like a fish asking what is the use of water, or a person what is the use of air.
There is a common confusion about meaning versus application. Trigonometry has no meaning per se, but it has lots of application. Strictly speaking, trigonometry, like all math, is a made-up system of logically consistent axioms and definitions. Its usefulness is a completely separate matter. Of course historically it was made up intentionally to be useful, but it makes no more sense to ask what the meaning of trigonometry is than to ask what the meaning of "e-r" verbs in French is. They have no meaning of their own - they have definitions, but that's not the kind of meaning I'm describing. To say that "jouer" means "to play" is like saying tan(theta) means sin(theta)/cos(theta), or y/x for any point on the unit circle, or any other definition one chooses to use. But "what does tan(theta) mean?" is itself a meaningless question. It's a great question, because it reveals something deep about the nature of mathematics, but ultimately it is meaningless. In other words, we need to ask different questions.
I tell students that math is like a game. We make up the rules in a way that works - i.e. we can't have 2 rules that contradict each other or we wouldn't be able to play the game. Then we just play the game and see what happens. There's no talk about justifying doing this, or the usefulness of it, any more than there is in a language or music class. If you don't want to do it, then don't do it. If you're in a degree program that requires you to learn math, then presumably you want to do it, even if only to get the degree.
Memory devices like this hexagon are great for remembering math facts, however the downside is that they can obscure the big picture, the relationships between mathematical objects. For example, if someone asks "why is tangent equal to sine over cosine?", the answer is not, "because if you go clockwise around the hexagon..." Ironically, what most students want to know when they talk about meaning, is the "why", which is great because that is indeed the important question. But unfortunately they expect the answer to be physical, to have something to do with the "real world", and their commitment to a "practical" explanation is what oftentimes keeps them from being open to the real meaning, the true "why". That meaning is in the structure, in the relationships between mathematical objects, and this is what math is really about; not numbers or measurements or calculations.
Too many students use memory devices as an excuse to not learn the definitions of the trig functions, or how to derive the pythagorean identities, or to prove the pythagorean theorem. I would argue that a number-crunching, algorithmic approach to math - even full of applications - is the main culprit of the bad reputation math has among apathetic students. Math is very useful for solving problems in science and engineering, but those problems are not math strictly speaking. We don't wait to start doing "real math" until we've done years of "classroom math"; that classroom math is the real math. We start on the very first day, and never stop. That other stuff is engineering and business and science - wonderful and useful endeavors that require math to do, but not math itself. That would be like saying that you have to study "classroom French" for years before you can do "real French" by writing a business contract in French. Give the contract to a native-French speaker and ask them what it is, and (assuming they recognize it) they'd say it's a contract, not "French", "real" or otherwise. Likewise, ask a physicist what Maxwell's Equations are, and they'd say "the foundational description of classical electromagnetism" or something to that effect; they wouldn't say they are math - even though that would be true - they are differential equations, which is math. That would be just as strange as saying that Moby Dick, or the U.S. Constitution, is English. The flipside is that if you don't speak English, you can't read Moby Dick or the U.S. Constitution; and if you can't "read" differential equations, you can't do physics.
Interestingly, I've never had a multi-lingual student ask me when they would use math outside a classroom. I suspect that the experience of learning a foreign language (and in some cases, several) has revealed the strangeness of such a question.
Larry David You HAVE to answer this question, not doing so is bad teaching.
Because nobody is going to memorize this stuff past the exam then.
Language is obvious, but math isn‘t, ofc it isn‘t with these stupid ladder problems in trig, or quadratic equations with basketballs.
How would you get behind the real usage of this stuff? Or do you seriously think trig is used for calculating right ladder angles?
You have to pick good, interesting problems, always focus on real world usage.
Our teacher did this, as much as the rules allowed him to, and this is also the stuff me & my former classmates best remember.
osm! TX
👍🏻
I use piattos chips to visualize this from my stdents
That sounds interesting.
this is worst than basic concept
It definitely requires a little work to memorize the various patterns, but regardless, memorization is required no matter what mnemonic device is used. Hope you find a method that works for you, since this one did not.
Pods punda
Thanks
You're welcome! Thanks for watching.