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Given ${\displaystyle \cot \left(b\right)=-2}$ find ${\displaystyle \sin \left(4b\right)}$ and ${\displaystyle \cos \left(4b\right)}$
I made $\sin \left(4b\right)$ into a expanded form.
${\displaystyle 4\sin \left(b\right){\cos }^3\left(b\right)-4{\sin }^3\left(b\right)\cos \left(b\right)}$ And then I made a triangle using ${\displaystyle \cot \left(b\right)=-2}$for information.
I got from that triangle that, ${\displaystyle \sin \left(b\right)=\frac{\sqrt{5}}{5}}$ and that ${\displaystyle \cos \left(b\right)=\frac{-2\sqrt{5}}{5}}$
And from their I simplified.
But whenever I expand ${\displaystyle \cos 4b}$ and plug in I get a bad answer, ${\displaystyle \frac{353}{25}}$ and I do not know if it is correct or wrong.

Assessing stability or instability of a system of equations with complex eigenvalues
Having this system , we get clearly two complex eigenvalues. If one has to assess the stability of the system at these eigenvalues, we have for the matrix A:
$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$
that ${\displaystyle T^2-4\mathrm{\Delta }\ge \text{or}\le \text{or}=0}$, where ${\displaystyle T=a+c}$, while ${\displaystyle \mathrm{\Delta }=ad-bc=DetA}$. But with complex eigenvalues ${\displaystyle \pm 2i}$, T which must be real, becomes complex and ${\displaystyle \delta }$ is always 0, when it would vary from greater than or lesser than zero for real eigenvalues. How do we solve this with complex eigenvalues?

Heat transfer between two fluids through a sandwiched solid (coupled problem):
Two fluids ${\displaystyle (t_h,t_c)}$ flow opposite to each other on either side of a solid (T), while exchanging heat among themselves. In such a scenario, the conduction in the solid is governed by:
${\displaystyle x\in [0,1],y\in [0,1]}$
1) $\kappa \frac{{\mathrm{d}}^{2}T}{\mathrm{d}{x}^2}+\mu b_h(t_h-T)-\nu b_c(T-t_c)=0$
with boundary condition as ${\displaystyle T^{\prime }\left(0\right)=T^{\prime }\left(1\right)=0}$
The fluids are governed by the following equations:
2) $\frac{\mathrm{d}{t}_h}{\mathrm{d}x}+b_h(t_h-T)=0$
3) $\frac{\mathrm{d}{t}_c}{\mathrm{d}x}+b_c(T-t_c)=0$
The hot fluid initiates at ${\displaystyle x=0}$ and the cold fluid starts from ${\displaystyle x=1}$.
The boundary conditions are ${\displaystyle t_h(x=0)=1}$ and ${\displaystyle t_c(x=1)=0}$
Equation (1), (2) and (3) form a coupled system of ordinary differential equations.
It is pretty evident that using (2) and (3), Equation (1) can be re-written as:
4) $\kappa \frac{{\mathrm{d}}^{2}T}{\mathrm{d}{x}^2}-\mu \frac{\mathrm{d}{t}_h}{\mathrm{d}x}+\nu \frac{\mathrm{d}{t}_c}{\mathrm{d}x}=0$
However, I have not been able to proceed further.
Some parameter values are
${\displaystyle b_c=12.38,b_h=25.32,\mu =1.143,\nu =1,\kappa =2.16}$

find the Fourier transform of:  f(t) = cos(3*pi*t)[u(t+4) - u(t-4)] :

a) using  using the Fourier definition the integral identity

b) Finalize your answer using the sinc function

Write a quadratic equation in factored form that has solutions at 6 and -⅔ 

determine the lowest common multiple of the following 4x2y2, 6xy, 10xy2