Recent content by chwala

With complex numbers, it makes it even easier,...

done.

for part i. i guess the quadratic equation was not indicated. I will go ahead and finish on that; The quadratic equation will be ##x^2 -p(p^2+3c)x^2-c^3=0##

a different line for ##\tan 4S## ##\tan (4S)= \tan (2S +2S) = \dfrac{\tan 2S + \tan 2S}{1 - \tan^2 (2S)}## ##\tan (2S) = \dfrac{5}{12}## therefore, ##\tan 4S= \left[ \dfrac{\dfrac{5}{12} + \dfrac{5}{12}}{1-\left[\dfrac{5}{12}\right]^2}\right]## ##\tan 4S=...

I note that in my proof, i did not have the ##4S## and instead worked with ##S## ... that was wrong. Your working clearly shows that error on my part. Cheers.

Noted, @berkeman. Is there a way to establish an award in the area of expertise of those profound members who have passed away? or a dedicated page... I believe that would be a true tribute to the fallen soldiers. It's sad that when someone dies, they are quickly forgotten; such is the nature of...

My condolences to the family. Could we at least have a picture of him during his happy days? That is the only way his memory can live on within us. Forever and ever! My thoughts and blessings!

My special thanks to Fresh42, Mark44, Bvu, Perok, SammyS, Haruspex, Orodruin, Chestermiller, WWGD, Martinbn, and the other person from South Africa for the elaborate math discussions over the years. I look forward to a more productive 2026 and beyond! Long live Physics Forums!

Thks @Gavran ,I will check on your approach...

This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0##...

@Mark i re-uploaded, i too could not see the highlighted part :biggrin:

Thanks for the thoughtful input — actually, I’ve taken this as a personal challenge! I’ll be applying the Runge-Kutta method to this problem and working through it manually as far as possible. I’m also committed to generating more of these hybrid-style problems that push the boundaries of how we...