User:Philc 0780/Sandbox2

resoance proj
$$V = IR\,$$ $$V = Q\frac{1}{C}$$ $$V = L\frac{dI}{dt}$$

$$V = R\frac{dQ}{dt}\,$$ $$V = Q\frac{1}{C}$$ $$V = L\frac{d^2Q}{dt^2}$$

$$E(t) = Q\frac{1}{C} + R\frac{dQ}{dt} + L\frac{d^2Q}{dt^2}$$ $$\dot{E}(t) = \frac{1}{C}I + R\dot{I} + L\ddot{I}$$

$$E(t) = E_0\cos(\omega t)\,$$

$$I = \alpha \cos(\omega t) + \beta \sin(\omega t)\,$$ $$\dot{I} = -\alpha \omega \sin(\omega t) + \beta \omega \cos(\omega t)\,$$ $$\ddot{I} = -\alpha \omega^2 \cos(\omega t) -\beta \omega^2 \sin(\omega t)\,$$

coeffsin

$$-E_o \omega = -L \omega^2 \beta - R \omega \alpha + \frac{\beta}{C}$$

coeffcos

$$0 = -L \omega^2 \alpha - R \omega \beta + \frac{\alpha}{C}$$

$$I = \frac{E_0R}{R^2+(L\omega - \frac{1}{\omega C})^2} \cos(\omega t) + \frac{E_0(L\omega - \frac{1}{\omega C})}{R^2+(L\omega - \frac{1}{\omega C})^2} \sin(\omega t)$$

$$I = \frac{E_0}{R^2+(L\omega - \frac{1}{\omega C})^2} \left ( R\cos(\omega t) + (L\omega - \frac{1}{\omega C}) \sin(\omega t) \right ) $$

$$I = \frac{E_0}{R^2+(L\omega - \frac{1}{\omega C})^2} \sqrt{R^2+(L\omega - \frac{1}{\omega C})^2}\cos(\omega t + \delta) $$

$$I = \frac{E_0\cos(\omega t + \delta)}{\sqrt{R^2+(L\omega - \frac{1}{\omega C})^2}},\quad \delta = \arctan \left (\frac{L \omega^2 C - 1}{R \omega C} \right ) $$

$$V = \sqrt{V_R^2 + (V_L - V_C)^2}$$

$$V = I\sqrt{R^2 + (L \omega - \frac{1}{\omega C})^2}$$

$$Z = \sqrt{R^2 + (L \omega - \frac{1}{\omega C})^2}$$

$$tan \delta = \frac{V_L - V_C}{V_R} = \frac{L \omega - \frac{1}{\omega C}}{R}$$

$$L \omega - \frac{1}{\omega C}=0$$ $$\omega = \frac{1}{\sqrt{LC}}$$

trig
$$\cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} \cos{(n-2k)\theta}, \quad \mbox{if }n\mbox{ is odd}$$

$$\cos^n\theta = \frac{1}{2^n} \binom{n}{\frac{n}{2}} + \frac{2}{2^n} \sum_{k=0}^{\frac{n}{2}-1} \binom{n}{k} \cos{(n-2k)\theta}, \quad \mbox{if }n\mbox{ is even}$$

$$\sin^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} (-1)^{(\frac{n-1}{2}-k)} \binom{n}{k} \sin{(n-2k)\theta}, \quad \mbox{if }n\mbox{ is odd}$$

$$\sin^n\theta = \frac{1}{2^n} \binom{n}{\frac{n}{2}} + \frac{2}{2^n} \sum_{k=0}^{\frac{n}{2}-1} (-1)^{(\frac{n}{2}-k)} \binom{n}{k} \cos{(n-2k)\theta}, \quad \mbox{if }n\mbox{ is even}$$