At 3:31, it should say -60 degrees, NOT -30 degrees. I sincerely apologize for such an elementary mistake and possible confusion. Thank you so much, fadhil rafi, for notifying me! =)
Sir I don't have so big heart as to give love to your channel ...you with your is great...Salute Sir!😃 And thank-you for you increased frequency of making these videos
I see the title of the video and I automatically think "I give up" because trig is one of the things my high school skips over, especially inverse trig.
I did (5+i)(4+i)(3+i)X=1+i X(a+bi)=(1+i)/(48+46i) You will get a result in a+bi form, (divide a/b and you will get 47) I just want to make sure if this way of working this question out is the right way. Edit: (94+2i)/(48^2+46^2) Therefore, a/b=47. If anybody finds any problem with it please let me know.
Wow! That is simply ingenious! I did not think of multiplying well-chosen complex numbers and finding the missing angle by taking advantage of the complex multiplication's geometric property. Again, wow! I am impressed. I do not see any errors in your argument. I just re-worked the problem (using your method), and I did indeed get 47. Fantastic job! =)
Of course, I may not be the best person to post a full explanation because I did not come up with it myself; however, I will give it a try. First, realize that 5+i has angle measuring arctan(1/5) in polar form. If this is unclear, try plotting 5+i in the complex plane--the angle the segment to 5+i makes with the x axis has tangent of 1/5. Similarly, 4+i, 3+i, and 1+i have angles measuring arctan(1/4), arctan(1/3), and arctan(1), respectively. Now note that when you multiply two complex numbers with angles a and b in the complex plane, the product has the angle a PLUS b (if you are not aware of this fact, try Googling multiplying complex numbers in polar form); that is, the angles add up when you multiply bunch of complex numbers. Thus, the complex number (5+i)(4+i)(3+i) has the angle measuring arctan(1/3)+arctan(1/4)+arctan(1/5), which is just what we want (realize that we don't have to worry about absolute values of the complex numbers because they are independent of angles). Now, we have the equation (5+i)(4+i)(3+i)(a+bi)= 1+i. Technically, we want to find d+i such that a+bi can be written as c(d+i) for some real c. However, note that d+i and a+bi both have the same angle (since they are constant multiples of each other), so all we have to find is the tangent of the angle measure for a+bi, or b/a. Solving above equation for a+bi, we get a+bi = (1+i)/(48+46i), or (1+i)(48-46i)/(something real). We can discard (something real) part because that does not affect the angle. Finally, (1+i)(48-46i) = 94+2i, whose angle's tangent is 2/94 = 1/47. Thus, our final answer is 94/2 = 47. If enough viewers wish me to make a video formally explaining this in-depth, I am more than willing to.
this is long question and it is not very hard if u know the limit of arctan and tan x+y since it is add and faction the best way to caculate n is tan x+y
Arctangents are angles. Adding a bunch of angles is the same as multiplying a bunch of complex numbers. The number 3 + i has an angle of arctan 1/3, 4 + i has an angle of arctan 1/4, etc. And pi/4 is the arctan of 1, which is arctan of A/A. So all you need to solve is (3+i)(4+i)(5+i)(n+i) = A + Ai. Whatever your result is on the left, set the real and imaginary parts equal to each other and solve for n. Easy!
At 3:31, it should say -60 degrees, NOT -30 degrees. I sincerely apologize for such an elementary mistake and possible confusion. Thank you so much, fadhil rafi, for notifying me! =)
Sir I don't have so big heart as to give love to your channel ...you with your is great...Salute Sir!😃 And thank-you for you increased frequency of making these videos
Ce sont de très bons exercices, merci beaucoup pour toutes tes explications détaillées!
I see the title of the video and I automatically think "I give up" because trig is one of the things my high school skips over, especially inverse trig.
I did (5+i)(4+i)(3+i)X=1+i
X(a+bi)=(1+i)/(48+46i)
You will get a result in a+bi form, (divide a/b and you will get 47)
I just want to make sure if this way of working this question out is the right way.
Edit: (94+2i)/(48^2+46^2)
Therefore, a/b=47.
If anybody finds any problem with it please let me know.
Wow! That is simply ingenious! I did not think of multiplying well-chosen complex numbers and finding the missing angle by taking advantage of the complex multiplication's geometric property. Again, wow! I am impressed. I do not see any errors in your argument. I just re-worked the problem (using your method), and I did indeed get 47. Fantastic job! =)
Can you explain this a bit? Where do those complex factors come from?
Yes i also want to know!
Of course, I may not be the best person to post a full explanation because I did not come up with it myself; however, I will give it a try.
First, realize that 5+i has angle measuring arctan(1/5) in polar form. If this is unclear, try plotting 5+i in the complex plane--the angle the segment to 5+i makes with the x axis has tangent of 1/5. Similarly, 4+i, 3+i, and 1+i have angles measuring arctan(1/4), arctan(1/3), and arctan(1), respectively.
Now note that when you multiply two complex numbers with angles a and b in the complex plane, the product has the angle a PLUS b (if you are not aware of this fact, try Googling multiplying complex numbers in polar form); that is, the angles add up when you multiply bunch of complex numbers. Thus, the complex number (5+i)(4+i)(3+i) has the angle measuring arctan(1/3)+arctan(1/4)+arctan(1/5), which is just what we want (realize that we don't have to worry about absolute values of the complex numbers because they are independent of angles).
Now, we have the equation (5+i)(4+i)(3+i)(a+bi)= 1+i. Technically, we want to find d+i such that a+bi can be written as c(d+i) for some real c. However, note that d+i and a+bi both have the same angle (since they are constant multiples of each other), so all we have to find is the tangent of the angle measure for a+bi, or b/a.
Solving above equation for a+bi, we get a+bi = (1+i)/(48+46i), or (1+i)(48-46i)/(something real). We can discard (something real) part because that does not affect the angle. Finally, (1+i)(48-46i) = 94+2i, whose angle's tangent is 2/94 = 1/47. Thus, our final answer is 94/2 = 47.
If enough viewers wish me to make a video formally explaining this in-depth, I am more than willing to.
LetsSolveMathProblems no thanks got this explanation 😊
I was REALLY hoping that n would equal 6 for continuity’s sake.
Still a great video though!
1/n = tan(π/4-arctan(1/3)-arctan(1/4)-arctan(1/5))
Lazy method.
Johannes H They don't have calculators , right?
No calculators (or any other electronic devices) are allowed on AIME.
this is long question and it is not very hard if u know the limit of arctan and tan x+y
since it is add and faction the best way to caculate n is tan x+y
Well, that's an argument 😁
@@morgard211, don't need calculators. Evaluate tan(A), tan(A-B), tan(A-B-C) and tan(A-B-C-D) and voilà! Note: A=π/4, B=arctan(⅓), etc.
I have no idea what arctan is but it seems interesting.
Simply excellent 👍
Thank you, San Seng! =)
Arctangents are angles. Adding a bunch of angles is the same as multiplying a bunch of complex numbers. The number 3 + i has an angle of arctan 1/3, 4 + i has an angle of arctan 1/4, etc. And pi/4 is the arctan of 1, which is arctan of A/A. So all you need to solve is (3+i)(4+i)(5+i)(n+i) = A + Ai. Whatever your result is on the left, set the real and imaginary parts equal to each other and solve for n. Easy!
It would be very great Sir if you include som tougher problems of national level Olympiads like USAJMO
This problem was actually in AoPS's Precalculus textbook!
I did very similar to Ramesh: (3+i)(4+i)(5+i)(n+i)=Rcis45 and when simplify left side then x=y in the x+yi result
Like your voice
He solved that problem and all you have to say is "I like your voice"😂
Ark'tan, Ark'tan, Ark'tan... That's satisfied
sos un capo, segui asi
I solved this using complex numbers
i clicked video thinking u will show some better method to do this maybe a shorter method
How tf do you do this shit