Solve ... for a
0-3+3cos(a/3)=-1/(sin(a/3)) * (9-a)
... a is supposed to be a=5.41
Rounded to two places, 5.41 is one of the three solutions.
We can simplify the expressions above, by making a substitution.
Let k = a/3 (so a = 3*k)
This substitution leads to an equivalent equation:
1 - cos(k) = (3 - k)/sin(k)
Using an identity for sin(2*k) we also get:
sin(k) - 1/2*sin(2*k) = 3 - k
Graphing both sides of either of these equations shows three solutions for k (see below). Tripling each of these solutions yields the solutions for a.
When we have an equation comprised of a trigonometric function set equal to a polynomial function, it is often impossible to solve by hand.
I approximated the three solutions for k, by zooming in on the intersection points of a graph. One could also use a Computer Algebra System, to approximate the solutions.
There are methods for approximating by hand; have you heard of Newton's Method?