stickman lore Official š | 2048 / 0 to ????? Inf
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Helping everything
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stickman lore Official š | 2048 / 0 to ????? Inf
Arctanget calculator: (copy-paste)
https://www.desmos.com/calculator/5dvms4c5xb
2 weeks ago (edited) | [YT] | 0
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stickman lore Official š | 2048 / 0 to ????? Inf
My masterpiece
Sin(2x)=?
2x=x+x
sin(x+x)
Sin(a+b)=?
sin(a+b)
b=ci
sin(a+ci)
(e^(a+ci)i-e^-(a+ci)i)/2i
(e^(-c+ai)-e^(c-ai))/2i
(e^-c * e^ai - e^c * e^-ai)/2i
Pt 1
e^-c * e^ai
e^-c * (cos(a)+sin(a)i)
cos(a)e^-c + isin(a)e^-c
Pt 2
e^c * e^-ai
e^c * (cos(a)-sin(a)i)
cos(a)e^c - isin(a)e^c
Pt 3
(cos(a)e^-c + isin(a)e^-c)-(cos(a)e^c - isin(a)e^c)
(cos(a)e^-c - cos(a)e^c)+ i(sin(a)e^-c + sin(a)e^c)
Pt 4
(cos(a)e^-c - cos(a)e^c) + i(sin(a)e^-c + sin(a)e^c) we want this to divide by 1/2i we know that 1/i=-i
Pr 5
i(cos(a)e^-c - cos(a)e^c) + iĀ²(sin(a)e^-c + sin(a)e^c)
(-sin(a)e^-c - sin(a)e^c) + i(cos(a)e^-c - cos(a)e^c)
Pt 6
-(-sin(a)e^-c - sin(a)e^c) - i(cos(a)e^-c - cos(a)e^c)
(sin(a)e^-c + sin(a)e^c) + i(cos(a)e^c - cos(a)e^-c)
Pt 7
(sin(a)e^-c + sin(a)e^c)/2 + i(cos(a)e^c - cos(a)e^-c)/2
Pt 8
sin(a)(e^-c + e^c)/2 + icos(a)(e^c - e^-c)/2
sin(a)cosh(c) + icos(a)sinh(c)
Pt 9
b=ci
b/i=c
c=-bi
sin(a)cosh(-bi) + icos(a)sinh(-bi)
sin(a)cosh(bi) - icos(a)sinh(bi)
sinh(bi)=(e^bi - e^-bi)/2=isin(b)
cosh(bi)=(e^bi + e^-bi)/2=cos(b)
sin(a)cos(b) - icos(a)isin(b)
sin(a)cos(b) + cos(a)sin(b)
Part 10
Subsitute a=x, b=x
sin(x)cos(x) + cos(x)sin(x)
Sin(2x)=2sin(x)cos(x)
1 month ago (edited) | [YT] | 1
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stickman lore Official š | 2048 / 0 to ????? Inf
Well in that case
2 months ago | [YT] | 0
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stickman lore Official š | 2048 / 0 to ????? Inf
Cbrt(x) of a complex numbers
DEEDE
Do some expansion
(a+bi)Ā³=aĀ³+3aĀ²bi-3abĀ²-bĀ³i=(aĀ³-3abĀ²)+(3aĀ²b-bĀ³)i
(aĀ³-3abĀ²)=c, (3aĀ²b-bĀ³)=d
(aĀ³-3abĀ²)=c
Lets act c as a constant
a(aĀ²-3bĀ²)=c
(aĀ²-3bĀ²)=c/a
(-3bĀ²)=c/a - aĀ²
(bĀ²)=(aĀ² - c/a)/3
(b)=sqrt((aĀ² - c/a)/3)
(3aĀ²sqrt((aĀ² - c/a)/3)-sqrt((aĀ² - c/a)/3)Ā³)=d
(3aĀ²sqrt((aĀ² - c/a)/3)-sqrt((aĀ² - c/a)/3)Ā³)=d
[(aĀ² - c/a)/3] * [(8aĀ² + c/a)Ā² / 9] = dĀ²
(aĀ²/3 - c/a) * (aĀ²/3 - c/a - 3aĀ²)Ā² = dĀ²
(aĀ²/3 - c/a) * (aĀ²/3 - c/a - 3aĀ²)Ā² = dĀ²
64aā¶/27 - 16aĀ³c/3 - 5cĀ² - cĀ³/aĀ³ = dĀ²
64aā¹/27 - 16aā¶c/3 - 5cĀ²aĀ³ - cĀ³ = dĀ²aĀ³
t=aĀ³
64tĀ³/27 - 16tĀ²c/3 - 5cĀ²t - t = dĀ²t
64tĀ³/27 - 16tĀ²c/3 - (5cĀ²-dĀ²)t - t =0
64tĀ³/27 - 16tĀ²c/3 - (5cĀ²-dĀ²-1)t=0
(64/27)xĀ³+(-9cĀ²+dĀ²+1)x+1/4(-23cĀ³+3cdĀ²+3c)
t=x+3c/4
(64/27)xĀ³+(-9cĀ²+dĀ²+1)x+1/4(-23cĀ³+3cdĀ²+3c)
x=Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)), (omegaCbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³))), (omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaCbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³))
t=((Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³))))-3c/4, (omegaCbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4, (omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaCbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4
a=cbrt(((Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³))))-3c/4), cbrt((omegaCbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4), cbrt((omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaCbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4)
(b)=sqrt(((cbrt(((Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³))))-3c/4))Ā² - c/(cbrt(((Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³))))-3c/4)))/3), sqrt(((cbrt((omegaCbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4))Ā² - c/(cbrt((omegaCbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4)))/3), sqrt(((cbrt((omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaCbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4))Ā² - c/(cbrt((omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaCbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4)))/3)
------------------------------
Cbrt((c,d))=((cbrt(((Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³))))-3c/4)),(sqrt(((cbrt(((Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³))))-3c/4))Ā² - c/(cbrt(((Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³))))-3c/4)))/3))), ((cbrt((omegaCbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4)),(sqrt(((cbrt((omegaCbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4))Ā² - c/(cbrt((omegaCbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4)))/3))), ((cbrt((omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaCbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4)),(sqrt(((cbrt((omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaCbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4))Ā² - c/(cbrt((omegaĀ²Cbrt((27/512)(23cĀ³-3cd-3c)+sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)) + omegaCbrt((27/512)(23cĀ³-3cd-3c)-sqrt(((27/512)(-23cĀ³+3cd+3c))Ā²+((9/64)(-9cĀ²+dĀ²+1))Ā³)))-3c/4)))/3)))
2 months ago | [YT] | 0
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stickman lore Official š | 2048 / 0 to ????? Inf
youtube.com/shorts/560agcaBVz...
Now āŖ@camman18ā¬ is forsaken
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stickman lore Official š | 2048 / 0 to ????? Inf
Linear
ax+b=0
ax=-b
x=-b/a
Quadradic
AxĀ²+bx+c
xĀ²+(b/a)x+(c/a)=0
Substitute x=y-h
A(y-h)Ā²+b(y-h)+c what h make linear coefficient 0 is b/2a
So we Substitute x=y-(b/2a)
This looks like something
(y-(b/2a))Ā²+(b/a)(y-(b/2a))+(c/a)
yĀ²+(4ac-bĀ²)/(4aĀ²)=0
yĀ²=-(4ac-bĀ²)/(4aĀ²)
y=Ā±sqrt(-(4ac-bĀ²)/(4aĀ²))
y=Ā±sqrt(-(4ac-bĀ²))/2a
y=sqrt((bĀ²-4ac))/2a
x=y-(b/2a)->x+(b/2a)=y
x+b/2a=Ā±sqrt((bĀ²-4ac))/2a
x=(-bĀ±sqrt((bĀ²-4ac)))/2a
Cubics
Cubic formula
axĀ³+bxĀ²+cx+d=0
x=y-b/3a
Expanding it gives us xĀ³+px+q for p=(c - bĀ²/3a), q=((2bĀ³-9abc+27aĀ²d)/27aĀ²)
xĀ³+px+q=0
xĀ³+px+q=0
(a+b)Ā³=aĀ³+3ab(a+b)+bĀ³
x=u+v
(u+v)Ā³+px+q=0
(a+b)Ā³-3ab(a+b)=aĀ³+bĀ³
(u+v)Ā³+p(u+v)+q=0
(u+v)Ā³+p(u+v)=-q
a=u, b=v
(a+b)Ā³+p(a+b)=-q
aĀ³+bĀ³=-q
3ab=p -> ab=p/3
ab=p/3
aĀ³bĀ³=pĀ³/27
K=aĀ³, L=bĀ³
K+L=-q, KL=pĀ³/27
(-q/2+B)+(-q/2-B)=-q
(-q/2+B)(-q/2-B)=pĀ³/27
-qĀ²/4 - BĀ²=pĀ³/27
BĀ² - qĀ²/4=-pĀ³/27
BĀ²=-pĀ³/27 + qĀ²/4
B=sqrt(-pĀ³/27 + qĀ²/4)
K=-q/2+sqrt(-pĀ³/27 + qĀ²/4)
L=-q/2-sqrt(-pĀ³/27 + qĀ²/4)
Quadradic formula form
K=(-q+sqrt(-4pĀ³/27 + qĀ²))/2
L=(-q-sqrt(-4pĀ³/27 + qĀ²))/2
And
aĀ³=(-q+sqrt(-4pĀ³/27 + qĀ²))/2
bĀ³=(-q-sqrt(-4pĀ³/27 + qĀ²))/2
For
a=cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)
b=cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2)
It thr
u=cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)
v=cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2)
frst root of depressed
x=cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)+cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2)
Ratonial root therom
Is xĀ²+cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)+cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2)x+cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)+cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2)=0
To (-(cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)+cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2))Ā±sqrt((cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)+cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2))Ā²-4(cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)+cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2))))/2
And
Final!!!
cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)+cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2)+b/3a
((-(cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)+cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2))Ā±sqrt((cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)+cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2))Ā²-4(cbrt((-q+sqrt(-4pĀ³/27 + qĀ²))/2)+cbrt((-q-sqrt(-4pĀ³/27 + qĀ²))/2))))/2)+b/3a
Into the!!!
cbrt((-((2bĀ³-9abc+27aĀ²d)/27aĀ²)+sqrt(-4(c - bĀ²/3a)Ā³/27 + qĀ²))/2)+cbrt((-((2bĀ³-9abc+27aĀ²d)/27aĀ²)-sqrt(-4(c - bĀ²/3a)Ā³/27 + ((2bĀ³-9abc+27aĀ²d)/27aĀ²)Ā²))/2)+b/3a
((-(cbrt((-((2bĀ³-9abc+27aĀ²d)/27aĀ²)+sqrt(-4(c - bĀ²/3a)Ā³/27 + ((2bĀ³-9abc+27aĀ²d)/27aĀ²)Ā²))/2)+cbrt((-((2bĀ³-9abc+27aĀ²d)/27aĀ²)-sqrt(-4(c - bĀ²/3a)Ā³/27 + ((2bĀ³-9abc+27aĀ²d)/27aĀ²)Ā²))/2))Ā±sqrt((cbrt((-((2bĀ³-9abc+27aĀ²d)/27aĀ²)+sqrt(-4(c - bĀ²/3a)Ā³/27 + ((2bĀ³-9abc+27aĀ²d)/27aĀ²)Ā²))/2)+cbrt((-((2bĀ³-9abc+27aĀ²d)/27aĀ²)-sqrt(-4(c - bĀ²/3a)Ā³/27 + ((2bĀ³-9abc+27aĀ²d)/27aĀ²)Ā²))/2))Ā²-4(cbrt((-((2bĀ³-9abc+27aĀ²d)/27aĀ²)+sqrt(-4(c - bĀ²/3a)Ā³/27 + ((2bĀ³-9abc+27aĀ²d)/27aĀ²)Ā²))/2)+cbrt((-((2bĀ³-9abc+27aĀ²d)/27aĀ²)-sqrt(-4(c - bĀ²/3a)Ā³/27 + ((2bĀ³-9abc+27aĀ²d)/27aĀ²)Ā²))/2))))/2)+b/3a
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stickman lore Official š | 2048 / 0 to ????? Inf
What method do you use
1:
Arcsin'(t)
Use Arcsin(t)=-iln(it+sqrt(1-tĀ²))
D/dx -iln(it+sqrt(1-tĀ²))
-i(D/dx ln(it+sqrt(1-tĀ²)))
-i((d/dx it+sqrt(1-tĀ²))/(it+sqrt(1-tĀ²)))
d/dx it+sqrt(1-tĀ²)
First term: i
Second term: (-2t)/2sqrt(1-tĀ²) -> (-t)/sqrt(1-tĀ²)
i + (-t)/sqrt(1-tĀ²)
-i((i + (-t)/sqrt(1-tĀ²))/(it+sqrt(1-tĀ²)))
-i((isqrt(1-tĀ²)/sqrt(1-tĀ²) + (-t)/sqrt(1-tĀ²))/(it+sqrt(1-tĀ²)))
-i((isqrt(1-tĀ²)/sqrt(1-tĀ²) + (-t)/sqrt(1-tĀ²))/(it+sqrt(1-tĀ²)))
-i(((isqrt(1-tĀ²)-t)/sqrt(1-tĀ²))/(it+sqrt(1-tĀ²)))
-i(((isqrt(1-tĀ²)-t)/sqrt(1-tĀ²))(it+sqrt(1-tĀ²)))
(((sqrt(1-tĀ²)+ti)/sqrt(1-tĀ²))(it+sqrt(1-tĀ²)))
(sqrt(1-tĀ²)+ti)/sqrt(1-tĀ²)(it+sqrt(1-tĀ²)))
1/sqrt(1-tĀ²)
2:
Use formula d/dx f^-1(x)
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stickman lore Official š | 2048 / 0 to ????? Inf
ķ“ķė°ģ“ķø
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stickman lore Official š | 2048 / 0 to ????? Inf
Quadradic
For quadradic roots a, b if can be factored in (x-a)(x-b) and expanding is (xĀ²-xb)-(ax-ab)
(xĀ²-bx)+(ab-ax)
(((xĀ²)+(ab))-(bx))-ax
(xĀ²)+(ab)-(b+a)x
xĀ²-(b+a)x+ab
However quadradic term is1
For axĀ²+bx+c=0
Idea
xĀ²+(b/a)x+(c/a)=0
x_1+x_2=b/a
x_1x_2=c/a
To make it easier
Subsitute d=b/a, e=c/a (not euler's number)
x_1+x_2=d
x_1x_2=e
(d/2-j)+(d/2+j)=d
x_1x_2=e
(d/2-j)(d/2+j)=e
(dĀ²/4-jĀ²)=e
(-jĀ²)=e - dĀ²/4
(jĀ²)=dĀ²/4 - e
(j)=sqrt(dĀ²/4 - e)
(d/2 - sqrt(dĀ²/4 - e))(d/2 + sqrt(dĀ²/4 - e))=d
(d/2 - sqrt(dĀ²/4 - e)), (d/2 + sqrt(dĀ²/4 - e)) is (d/2 Ā± sqrt(dĀ²/4 - e))
(b/2a Ā± sqrt(bĀ²/4aĀ² - c/a))
(b/2a Ā± sqrt(bĀ²/4aĀ² - 4ca/4aĀ²))
Quadradic formula usually has 4ac
(b/2a Ā± sqrt(bĀ²/4aĀ² - 4ac/4aĀ²))
(b/2a Ā± sqrt((bĀ² - 4ac)/4aĀ²))
(b/2a Ā± sqrt(bĀ² - 4ac)/sqrt(4aĀ²))
(b/2a Ā± sqrt(bĀ² - 4ac)/sqrt(4aĀ²))
d=-b/a it has - in linear terms
(-b/2a Ā± sqrt(bĀ²/4aĀ² - c/a))
(-b/2a Ā± sqrt(bĀ²/4aĀ² - 4ac/4aĀ²))
(-b/2a Ā± sqrt((bĀ² - 4ac)/4aĀ²))
(-b/2a Ā± (sqrt(bĀ² - 4ac)/2a))
(-b Ā± (sqrt(bĀ² - 4ac)))/2a
Sum of roots: b/a (from old d)
Product of roots: c/a
For bĀ²-4ac<0: both are complex
For bĀ²-4ac=0: 1 solution
For bĀ²-4ac>0: 2 real solution
3 months ago | [YT] | 0
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stickman lore Official š | 2048 / 0 to ????? Inf
Linear
ax+b=0
ax=-b
x=-b/a
Quadradic
AxĀ²+bx+c
xĀ²+(b/a)x+(c/a)=0
Substitute x=y-h
A(y-h)Ā²+b(y-h)+c what h make linear coefficient 0 is b/2a
So we Substitute x=y-(b/2a)
This looks like something
(y-(b/2a))Ā²+(b/a)(y-(b/2a))+(c/a)
yĀ²+(4ac-bĀ²)/(4aĀ²)=0
yĀ²=-(4ac-bĀ²)/(4aĀ²)
y=Ā±sqrt(-(4ac-bĀ²)/(4aĀ²))
y=Ā±sqrt(-(4ac-bĀ²))/2a
y=sqrt((bĀ²-4ac))/2a
x=y-(b/2a)->x+(b/2a)=y
x+b/2a=Ā±sqrt((bĀ²-4ac))/2a
x=(-bĀ±sqrt((bĀ²-4ac)))/2a
Cubics
axĀ³+bxĀ²+cx+d
xĀ³+(b/a)xĀ²+(c/a)x+(d/a)=0
A(x-h)Ā³+b(x-h)Ā²+c(x-h)+d what h make quadradic coefficient 0
b/3a
Plug h=b/3a is xĀ³+px+q (p=(3ac-bĀ²)/3aĀ² and q=(2bĀ³ ā 9abc +27aĀ²d)/27aĀ³)
xĀ³+px+q=0
x=u+v
(u+v)Ā³+p(u+v)+q=0
(u+v)Ā³=uĀ³+3uĀ²v+3uvĀ²+vĀ³
(u+v)Ā³=uĀ³+3uĀ²*v+3uv*v+vĀ³
(u+v)Ā³=uĀ³+v(3uĀ²+3uv)+vĀ³
(u+v)Ā³=uĀ³+3v(uĀ²+uv)+vĀ³
(u+v)Ā³=uĀ³+3uv(u+v)+vĀ³
uĀ³+3uv(u+v)+vĀ³+p(u+v)+q=0
uĀ³+vĀ³+3uv(u+v)+p(u+v)+q=0
uĀ³+vĀ³+(3uv+p)(u+v)+q=0
uv=-p/3
uĀ³+vĀ³+q=0
uĀ³+vĀ³=-q
uv=-p/3 and uĀ³+vĀ³=-q
uĀ³vĀ³=-pĀ³/27 and uĀ³+vĀ³=-q
U=uĀ³, V=vĀ³
UV=-pĀ³/27 and U+V=-q
tĀ²+qt-pĀ³/27
Soon
3 months ago (edited) | [YT] | 0
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