Hi Everyone! I hope you love that video, which is practice with implicit differentiation, inverse functions, trig identities, and much more, all in a short video! The style of my channel is to concisely deliver important math ideas, concepts and topics to you! Don't forget to like ✅, subscribe 🥇 and share ⏩ for *elite infinite free accessible math education worldwide* at *all levels of math*. Check out my popular video: th-cam.com/video/Fneg7KUxOhg/w-d-xo.htmlsi=V_yd2fHjc-nMX4jo to *master logarithmic differentiation* from beginner to master in 15 min! 🎁
So, I have a constant of pi/2. Found it by graphing arcsin(x) vs arccos(x) and seeing that that arccos(x) was basically the 'same' as arcsin, but inverted and shifted. Then graphing -arccos(x) - orientation was identical, but shifted by pi/2. That lead to arcsin(x) = -arccos(x) + pi/2 which reminds me of the cofunction identity sin(x) = cos(pi/2 - x) = -cos(x + pi/2). Either way, appreciate the videos.
Hi @cyanidepress thank you so much for your comment and for sharing your approach! 😊 Yes, that's a cool way of looking at it! 🥳 As you say, it follows from that trig identity sin(x) = cos(π/2 - x). Indeed, if y = sin(x), then arcsin(y) = x (by definition of arcsin) and arccos(y) = π/2 - x (by the identity and the definition of arccos). In particular, arcsin(y) + arccos(y) = x + (π/2 - x) = π/2 for all y. An alternate approach is: since we know arcsin(x) + arccos(x) is a constant by differentiation, to figure out the constant we just need to plug in one value of x. We can (for example) plug in x = 0 and obtain arcsin(0) + arccos(0) = 0 + π/2 = π/2. (We could have also used x = 1/2, x = 1/√2, x = √3/2, x = 1, since we readily know the inputs that give these outputs for both sin and cos.) However, I think the identity sin(x) = cos(π/2 - x) that you mention is indeed the best way to solve this problem since it doesn't use differentiation and allows you to immediately why arcsin(x) + arccos(x) = π/2. 😊 Thanks for your support of the videos and I hope you have an amazing day/evening/night! 😊
Hi @RiteshArora-s9x thanks so much for your comment! 😊 I'm going to be starting a real analysis lecture series on my channel very soon, and I hope that will be helpful (if you subscribe, you will be notified, but also it can be found on the "Playlists" tab on my channel). In terms of books for real analysis (1) "Principles of Mathematical Analysis" by Walter Rudin is an amazing textbook, it's challenging, but if you master it, then it will give you a really strong foundation (2) "Elementary Analysis: The Theory of Calculus" by Kenneth Ross is another wonderful book (I used it a bit myself at some point and I enjoyed reading it - it could be used together with Rudin's book and maybe a bit more leisurely to read) Abstract Algebra: (1) "Abstract Algebra" by Dummit and Foote is quite popular and comprehensive and leisurely to read (I think) - I haven't used it but I know it is very popular I'd recommend looking at those and seeing how you go and also please check out my upcoming real analysis lectures if you are interested in further supplements! I will be beginning with the theory of sequences and convergence and a video will be out in a few days! 🥳 I hope you have an amazing day/evening/night! 😊
Hi @uffe997 thank you so much for sharing! 😊 The constant is π/2 because if we plug in one value (say x = 0) into arcsin(x) + arccos(x), then we obtain arccos(0) + arcsin(0) = π/2 + 0 = π/2. The key point is that arccos(0) = π/2 since cos(π/2) = 0 and arcsin(0) = 0 since sin(0) = 0 (by definition of the inverse function). (If you plugged in any other value, you would also get π/2 because the derivative of arcsin(x) + arccos(x) is 0, but with specific values like 0, 1/√2, 1/2 √3/2, 1 etc. you can actually work out the individual terms and see this in action.) Thanks so much for watching, commenting and sharing your thoughts! 😊 I hope you have an amazing day/evening/night! 😊
Hi Everyone! I hope you love that video, which is practice with implicit differentiation, inverse functions, trig identities, and much more, all in a short video! The style of my channel is to concisely deliver important math ideas, concepts and topics to you! Don't forget to like ✅, subscribe 🥇 and share ⏩ for *elite infinite free accessible math education worldwide* at *all levels of math*. Check out my popular video: th-cam.com/video/Fneg7KUxOhg/w-d-xo.htmlsi=V_yd2fHjc-nMX4jo to *master logarithmic differentiation* from beginner to master in 15 min! 🎁
I loved it.
Thank you so much!!! 😊
So, I have a constant of pi/2. Found it by graphing arcsin(x) vs arccos(x) and seeing that that arccos(x) was basically the 'same' as arcsin, but inverted and shifted. Then graphing -arccos(x) - orientation was identical, but shifted by pi/2. That lead to arcsin(x) = -arccos(x) + pi/2 which reminds me of the cofunction identity sin(x) = cos(pi/2 - x) = -cos(x + pi/2). Either way, appreciate the videos.
Hi @cyanidepress thank you so much for your comment and for sharing your approach! 😊 Yes, that's a cool way of looking at it! 🥳 As you say, it follows from that trig identity sin(x) = cos(π/2 - x). Indeed, if y = sin(x), then arcsin(y) = x (by definition of arcsin) and arccos(y) = π/2 - x (by the identity and the definition of arccos). In particular, arcsin(y) + arccos(y) = x + (π/2 - x) = π/2 for all y.
An alternate approach is: since we know arcsin(x) + arccos(x) is a constant by differentiation, to figure out the constant we just need to plug in one value of x. We can (for example) plug in x = 0 and obtain arcsin(0) + arccos(0) = 0 + π/2 = π/2. (We could have also used x = 1/2, x = 1/√2, x = √3/2, x = 1, since we readily know the inputs that give these outputs for both sin and cos.) However, I think the identity sin(x) = cos(π/2 - x) that you mention is indeed the best way to solve this problem since it doesn't use differentiation and allows you to immediately why arcsin(x) + arccos(x) = π/2. 😊
Thanks for your support of the videos and I hope you have an amazing day/evening/night! 😊
sir pls tell book for abstract algebra and real analysis for undergrad for self study ? without teacher. thanks
Hi @RiteshArora-s9x thanks so much for your comment! 😊 I'm going to be starting a real analysis lecture series on my channel very soon, and I hope that will be helpful (if you subscribe, you will be notified, but also it can be found on the "Playlists" tab on my channel). In terms of books for real analysis
(1) "Principles of Mathematical Analysis" by Walter Rudin is an amazing textbook, it's challenging, but if you master it, then it will give you a really strong foundation
(2) "Elementary Analysis: The Theory of Calculus" by Kenneth Ross is another wonderful book (I used it a bit myself at some point and I enjoyed reading it - it could be used together with Rudin's book and maybe a bit more leisurely to read)
Abstract Algebra:
(1) "Abstract Algebra" by Dummit and Foote is quite popular and comprehensive and leisurely to read (I think) - I haven't used it but I know it is very popular
I'd recommend looking at those and seeing how you go and also please check out my upcoming real analysis lectures if you are interested in further supplements! I will be beginning with the theory of sequences and convergence and a video will be out in a few days! 🥳 I hope you have an amazing day/evening/night! 😊
@MathMasterywithAmitesh thanks sir.
I think the constant you asked for is zero
Hi @uffe997 thank you so much for sharing! 😊 The constant is π/2 because if we plug in one value (say x = 0) into arcsin(x) + arccos(x), then we obtain arccos(0) + arcsin(0) = π/2 + 0 = π/2. The key point is that arccos(0) = π/2 since cos(π/2) = 0 and arcsin(0) = 0 since sin(0) = 0 (by definition of the inverse function). (If you plugged in any other value, you would also get π/2 because the derivative of arcsin(x) + arccos(x) is 0, but with specific values like 0, 1/√2, 1/2 √3/2, 1 etc. you can actually work out the individual terms and see this in action.) Thanks so much for watching, commenting and sharing your thoughts! 😊 I hope you have an amazing day/evening/night! 😊