Solve s=8asin^2psi/6

$$s = 8 a \sin ^ { 2 } \frac { \psi } { 6 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{s}{4\left(\cos(\frac{\psi }{3})-1\right)}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\psi =6\pi n_{1}\\a\in \mathrm{C}\text{, }&s=0\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\psi =6\pi n_{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$8a\left(\sin(\frac{\psi }{6})\right)^{2}=s$$

The equation is in standard form.

$$8\left(\sin(\frac{\psi }{6})\right)^{2}a=s$$

Divide both sides by 8\left(\sin(\frac{1}{6}\psi )\right)^{2}.

$$\frac{8\left(\sin(\frac{\psi }{6})\right)^{2}a}{8\left(\sin(\frac{\psi }{6})\right)^{2}}=\frac{s}{8\left(\sin(\frac{\psi }{6})\right)^{2}}$$

Dividing by 8\left(\sin(\frac{1}{6}\psi )\right)^{2} undoes the multiplication by 8\left(\sin(\frac{1}{6}\psi )\right)^{2}.

$$a=\frac{s}{8\left(\sin(\frac{\psi }{6})\right)^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{s}{8\left(\sin(\frac{\psi }{6})\right)^{2}}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\psi =6\pi n_{1}\\a\in \mathrm{R}\text{, }&s=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\psi =6\pi n_{1}\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$8a\left(\sin(\frac{\psi }{6})\right)^{2}=s$$

The equation is in standard form.

$$8\left(\sin(\frac{\psi }{6})\right)^{2}a=s$$

Divide both sides by 8\left(\sin(\frac{1}{6}\psi )\right)^{2}.

$$\frac{8\left(\sin(\frac{\psi }{6})\right)^{2}a}{8\left(\sin(\frac{\psi }{6})\right)^{2}}=\frac{s}{8\left(\sin(\frac{\psi }{6})\right)^{2}}$$

Dividing by 8\left(\sin(\frac{1}{6}\psi )\right)^{2} undoes the multiplication by 8\left(\sin(\frac{1}{6}\psi )\right)^{2}.

$$a=\frac{s}{8\left(\sin(\frac{\psi }{6})\right)^{2}}$$

Solve for s

$$s=8a\left(\sin(\frac{\psi }{6})\right)^{2}$$