Solve 2*pi*R*k*sin(a) - 2*pi*a*k = m*g*tan(a) / (2*pi) for k.

Step-by-step solution

1. 1

   Spot that $k$ appears in both terms on the left. Neither side needs expanding; the whole problem is a factorisation. Both left-hand terms contain the common factor $2\pi k$: $2\pi Rk\sin\alpha-2\pi ak=2\pi k\left(R\sin\alpha-a\right).$ Collecting $k$ into a single factor is what makes it possible to isolate it in one division.

2. 2

   Rewrite the equation with $k$ collected. $2\pi k\left(R\sin\alpha-a\right)=\frac{mg\tan\alpha}{2\pi}$ Everything multiplying $k$ now sits in one bracket, and everything else is on the right.

3. 3

   Divide by the whole coefficient of $k$. Dividing both sides by $2\pi\left(R\sin\alpha-a\right)$ gives $k=\frac{\dfrac{mg\tan\alpha}{2\pi}}{2\pi\left(R\sin\alpha-a\right)}.$ Note that the $2\pi$ already in the denominator on the right does not cancel the new one - dividing by $2\pi$ a second time multiplies the denominators together.

4. 4

   Simplify the double fraction. Multiplying the two denominators, $k=\frac{mg\tan\alpha}{4\pi^{2}\left(R\sin\alpha-a\right)}.$ The $4\pi^{2}$ is the step people most often get wrong, writing $2\pi$ or $2\pi^{2}$ instead.

5. 5

   State the restriction. The division is only valid when $R\sin\alpha-a\neq 0,$ i.e. $\sin\alpha\neq a/R$. If $R\sin\alpha=a$ the left side is identically zero, so the equation has no solution for $k$ unless $mg\tan\alpha$ is also zero, in which case every $k$ works.

6. 6

   Check the rearrangement with sample numbers. Take $R=2$, $a=0.5$, $\alpha=30^\circ$, $m=1$, $g=9.8$. The formula gives $k=\frac{9.8\times 0.577350269}{4\pi^{2}(2\times 0.5-0.5)}=0.286639282.$ Substituting back into the original left side: $2\pi(2)(0.286639282)(0.5)-2\pi(0.5)(0.286639282)=0.900503863$, and the right side is $\dfrac{9.8\times 0.577350269}{2\pi}=0.900503863$. The two agree, confirming the algebra.