t-Formulae ~ 03 ~ Used to Evaluate Certain Integrals
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- เผยแพร่เมื่อ 6 ต.ค. 2024
- One of my subscribers asked me to explain what t-formulae are, and how they are used in trigonometry. This is the last of three videos that I created in response to that request (thank you, Kristyn).
Because t-formulae allow us to convert trigonometric functions into fractions involving polynomials, we can use them to change certain trigonometric integrals to 'integrals of fractions containing polynomials.' This can often (but not always) be an advantage.
Such substitutions are particularly valuable when the original integral only involves single instances of trigonometric functions and no powers. In fact, this is when t-formulae are almost exclusively used. In other words, we look for integrals that involve uses of sin(x), cos(x), and tan(x), but not powers of these functions.
The fraction that results will contain two polynomials. These may produce a logarithmic structure, or may require a division to take place, or may lead to the use of partial fractions, or may even produce inverse tangent functions, for example ... but many of these structures are almost instantly recognisable and experienced mathematicians know how to 'handle' them.
In this video, I provide a 'typical' example ... ∫1/(3 + 5cosx).dx ... which is Jim Coroneos' 30th integral (see my TH-cam playlist, www.youtube.co....
Because the complete solution of this integral has been accomplished in another video, I only spend sufficient time with the integral to show how the use of t-formulae allows us to convert it to a much more useable and recognisable form.
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Best wishes for your study and your mathematics!
Great example that clearly shows how helpful t-Formulae could be ! In this particular example there was a small nuisance od finding dx in order to use t-Formulae substitution, but after that everything develops quite nicely. Thank you, Graeme, for a valuable demonstration!
+MrVoayer Thank you for your faithful feedback, Mr Voayer.
Even the finding of dx develops some useful skills and, with practice, shoudl take no more than a few seconds. In fact, after a while, students would simply recognise it and know the conversion as a "fourth t-formula."
I send my warm wishes and greetings to you and your family, friend.
Whenever I am not sure about something in maths, I head over to your channel 😊
That comment is very kind of you, Nati. Thank you.
I am glad that my explanations help you!
Warm regards from south of Sydney.
Hi Graeme!
How are you nowadays ?
Happy new year 2021 !
Still remember how inspiring were (still are !) your "How to Study" series.
I watched them in my middle school and still is helping me when I have passed through my high school and my university (almost).
Thank you so much Graeme!
Hello Akash and 'Happy New Year' to you, too!
I am glad that my 'How to Study' videos have been of use to you ... and greatly appreciate your leaving such a warm comment to let me know. Thank you.
Very best wishes for your continuing studies!
Graeme
I came across our channel while looking to learn some concepts and you were the only one that had had it. Your content is so helpful.
Also Sir when is your next video coming out? Hope you in good health? I'm really anxious as you haven't posted in a long time (minimum 1 year), but you have replied to many comments. Please do start posting again.
I am glad that my videos have been helpful to you, Arkan.
It has actually been 4-5 years since I posted a video. During that time I have been dealing with many crises. I am hoping to resume creating videos within a year but have significant matters to attend to before then. I am sorry for the delay. As you notice, however, I have been able to respond to comments.
Kind regards to you from Australia :-)
@@CrystalClearMaths ok
Your tutorials are excellent. I love your channels, excellent teaching techniqiues, it help me to see and learn concept clearly.
Thank you, carvelbell181.
I am glad that my videos help you 'see and learn concepts clearly.' That is a great encouragement to me.
May you always continue to enjoy learning. Warm wishes to you.
Graeme
Hello sir how are you doing this is vikhyat here you were the only one who cleared my doubts of integrals i am now in a college doing well and how about you i hope you are too, and i hope you will be like this
Hi Graeme,
I was fiddling randomly on the internet and found your videos. I must tell you that I've become a fan of yours. I hope to see your next upload soon.
Regards,
Abhinav
Thank you very much, Abhinav.
Unfortunately, it may be close to another 12 months before I resume posting videos. I have 200-300 in the planning stage, but there have been (and are) a series of events/situations that are making the posting of videos extremely difficult for the moment.
Thank you for subscribing. I thank you for your patience and understanding ... and hope that you like the new videos when I am able to post them.
Kind regards,
Graeme
I'll happily wait for such quality stuff. Your videos deserve way more views. Much better than other channels. And I'm sorry to know about your parents and your situation. I wish you the very best.
@@AbhinavKrishna9991 ~ Thank you, kindly, Abhinav. Hopefully, you will not have to wait too long.
Sir pls come back 😭😭 we miss you
Thank you, friend.
I am working hard to get things in order to return, but we have had some major delays due to our government's reaction to the Covid situation. I was planning to resume in October this year, but it may be December before I can start producing videos again.
At least, after all these years, we are within months of producing material again.
Thank you for your encouragement and your patience.
Kind regards to you.
Great to see how many comments there are here recently paying homage Graeme!
I'm Harry, the young lad who illegally drove a car to one of your maths lessons when he was 15 because he was late.
Doing nothing related to maths today 10yrs later, but I remember on at least a weekly basis what you told me about persistence and work ethic, like your one-armed pullups
Thanks a load Graeme, hope you're very very well!
HB
Thank you for the videos sir. May I ask how you are doing these days?
Greetings, Abrar, and thank you very much for your concern.
My health has improved noticeably, but there is still a long way to go.
Because of Australian government lockdowns etc. my renovations have been delayed (they include a small studio) ... but I am still hoping to resume producing videos soon (in another month or two). It has been a long and frustrating journey.
Very kind regards to you.
Hi , i hope you are reading this , i want to tank you forward for all the immense efforts you have done to make maths more understandable , i wish u the best ,
Second , i want your help and opinion , i have my baccalauréat math exam tommorow , i had i would say inexperinced mistakes i have done through the year mainly i would say : that i start doubting my informations the day if the exams , i start noticing new things , careless mistakes , and a lack of concentration , if you have any solutions i can do regarding this i hope you provide , thank u !
Hello, Lazy Man. I am glad that you are finding mathematics more understandable, and that these videos are helping you along that journey.
I wish you the best for your baccalauréat examinations. With little time to go, the best thing that you can do for yourself is to get a good night's sleep, avoid eating sugars and lots of carbohydrates, avoid talking to other students before the exam (as they can add to your stress), and spend at least the first 20-30 minutes of the examination answering the very easiest questions. The last action helps your stress levels reduce so that you can think more clearly for the remainder of the exam.
You certainly need good sleep, otherwise you make too many careless errors.
I hope this helps and that you do the best you can.
Kind regards from Australia!
Are you alright bud? You plan on making any more vidoes?
Hello Aleksandar,
I'm sorry I haven't been posting for quite a while. I have been dealing with Complex-PTSD and trying to spend time with my dad who is dying of cancer as well as some other matters. Sadly, for the moment, I have had to put these videos and my website material "on the back burner."
I certainly plan on making more videos and have been drawing up plans for a large number of them but am not sure how soon I will be able to resume. Hopefully it will be within the next few weeks, but it could be another 2-4 months as we are not sure how much longer my father has to live.
Thank you so much for your concern, and for asking how things are going.
I trust all is well with you.
Warm regards,
Graeme
Glad to hear you are relatively alright, I can't begin to understand how it is but my mother suffers from PTSD from the Yugoslav wars. We moved away to a safe country but periodically she gets flashbacks. I hope your dads treatment doesn't cause too much pain.
I look forward to the videos, but don't feel rushed.
Aleks
Thank you very much for your concern and encouragement, Aleks.
I am so sorry to hear about your mother. All wars are brutal and living through one, as she did, leaves deep damage. I was spared that experience, but through my own journey I know that having PTSD is not easy! I hope she finds healing in time.
Thanks, too, for your thoughts about my father. It really is an issue of pain management at the moment and his pain has been very severe at times.
I hope you enjoy the videos when I start producing them again!
Warm regards to you from Australia.
Graeme
Hello Mr. Henderson, you have not made any videos in a while. What happened?
@@chocolatier9597 Hi Lea. I have just found your comment.
I an sorry that I have not been active for a while. There have been a number of major events in my life that have needed my attention. I am hoping that I will resume posting videos next year.
Thank you very much for your support and concern.
Warm regards.
Mr H
thank you so much for these videos you explain everything perfectly it's so incredibly helpful :)
You are very welcome, friend. I am glad that you find these videos helpful.
Hii sir
How are you sir?
Where have you gone ?
You didn't post any videos from so long
I thought you might be covering chapters like functions or limits or more calculus. You're an amazing professor and I would love to see more of your content
Thank you, kindly, Sahil.
I have been absent for a long time, it is true.
There have been a number of major family changes/calamities and some significant health issues as well, and we are trying to resolve the major issues so I can return to creating videos.
It has been a long journey.
I apologise for the delay, but assure you that I am trying to organise things as quickly as possible.
Thank you very much for your very kind remarks. They are appreciated!
Kind regards to you from Australia.
@@CrystalClearMaths ok sir
I hope everything goes well and you return soon. I pray for your better health and life . Thanks sir for helping students
@@sahilsharma6167 That is very kind of you, Sahil. Thank you very much for your prayers and for your kind thoughts. Both are greatly appreciated.
Happy 2020, I hope you are able to make more content this year.
Thank you, Socrates.
I wish you a happy 2020 also! I am hoping to have my workroom back (finally) and a few other things in place so I can resume creating content before the end of this year.
Warm regards.
I hope you come back. Your contribution is special.
Thank you very much, Socrates. It is getting closer.
There has been a huge amount going on which has kept me from producing videos etc. I would like to think that I will resume making videos next year.
Kind regards.
Also wishing you all the best and really hope you are doing well
I appreciate your kind thoughts, coffee beans ... very much.
We are coming out of a difficult few years, and I hope to resume creating videos around April or May next year.
Warm wishes to you, too!
Hope your doing well, Sir
I also hope so
Hi teacher really missed You and Your videos come back soon
Thank you for your kind thoughts. I am sorry that I have been away so long.
I hope to start posting videos again later this year ... perhaps around August.
There is much to do before then!
Kind regards to you from Australia.
@@CrystalClearMaths that sounds great be sure l will wait your return may God protect you❤️❤️❤️
@@ifmohammed6495 My goodness, you are fast! :-)
Thank you very much for your patience. God is, indeed, gracious.
Grace and peace to you, and I hope we will chat again soon.
@@ifmohammed6495 Thank you, If Mohammed.
@@CrystalClearMaths my pleasure actually I want to thank you for your ammazing explainsion I think you must have a website then alot of people will subscribe to your channel
why are you not making videos anymore :(
It is a very long story, Akif, with lots of drama and tragedy. Things are slowly improving and I hope to resume creating videos within the next few months.
I greatly appreciate your concern and your interest.
Kind regards to you.
@@CrystalClearMaths I would love to hear more about that
Its good to hear that you have not given up at teaching us
@@akifismail6597 Thank you, Akif.
It has been a journey of chronic illness and losing both my parents and a number of other illnesses and deaths in the family plus some other serious stresses. Fortunately, my health appears to be slowly improving. I am trying to get our home in order again so that I have somewhere to prepare and record videos.
You have been very patient, and I am grateful. Thank you.
@@CrystalClearMaths I feel so sorry for your losses
I hope you recover to have a happy life and start making yt videos if that makes you happy
Loosing both parents in a succession must be huge loss
I can pray for them through my own belief system
Good Luck and looking forward to this yt for more crystal clear maths 😉
@@akifismail6597 Thank you for your kind thoughts, Akif. They are appreciated.
Not entirely sure I understand what's going on in this video, mostly because I'm not at this level. But I've bookmarked for when I am so I Can come back :D
That is one of the great advantages of bookmarks, EssThree. I often bookmark videos that I wish to revisit so I can think about them more deeply.
I am not sure what 'level' of mathematics you are currently studying, but let me encourage you to become expert with the unit circle. The definitions of sine, cosine and tangent developed beyond ratios in right-angled triangles and were then linked to lengths related to the unit circle.
In NSW, Australia, you would need to study Extension Mathematics at senior level to learn about t-Formulae and angle sums and differences. If you have not yet encountered them and wish to learn about them, I suggest that you ask your teacher/lecturer.
I intend producing videos about this material, but have not been posting videos lately as I have been busy with other significant matters during the last few months. Hopefully, I will get to resume soon!
Ah true. I'm sort of muddling my way through math b at the moment in QLD. It was really good to see a video from an Australian on here haha
Good luck with your affairs, look forward to seeing more videos in the future :)
Thank you, ET :-). I grew up near the Queensland border and spent many holidays in Queensland until my late teens.
Among a number of other major things, my dad has just had major cancer surgery ... so the videos have been 'on hold' for a while. I will get back to them as soon as I can. I hope you like what you see.
Best wishes to you.
Sorry to hear that. Hope everything goes well!
Thank you very much, ET.
Hi!
I hope you are good and will come back :)
I am getting better and working hard to resume creating videos during the next few months, Speedcube Solver.
Thank you for your kind thoughts. They are encouraging :-)
@@CrystalClearMaths :)
@@speedcubesolver1195 ty :-)
Hi Sir how ru? I'm genuinely worried as you haven't posted in a long time. Are you fine?
Hi Lionel,
Thank you for your concern. I plan to produce a lot more material as soon as I can get to it, but a lot has been hampering my activity here. Among other things, I have significant PTSD, lost both my parents during the last year, am dealing with a couple of other major family matters, am renovating, and have my "studio" (for want of a better term) filled with material as a result of some of the above. I am trying to bet my health on track and empty out the studio so I can get to work again. Hopefully, you will not have to wait too long.
Very warm regards to you ... and thank you for your concern.
Graeme
hope you feel better mate!@@CrystalClearMaths
@@aviralverma5393 ~ Thank you very much, Lionel. We are making progress. Best wishes to you.
@@CrystalClearMaths are you getting better now?
@@OtiumAbscondita Hello Otium. I have improved significantly, thank you. There are still matters to deal with but I hoping to resume creating videos in the first half of next year ... hopefully, around March/April. Thank you very much for your care and concern.
hope you are doing well