There is no algebraic way to get a general solution when the
unknown appears both inside and outside of a so-called
'transcendental' function such as a trigonometric or logarithmic.
In general, such solutions can only be approximated by iterative
methods or infinite series.
The iterative method essentially is:
1. Try a number for an answer
2. Find that it doesn't work.
3. Find out if the answer tried is too big or too small.
4. If it's too big try one smaller, or if it's too small, try one bigger.
5. Repeat steps 3 and 4 and get as close as you like to an exact
solution, or until the approximate answer you get is close enough
for your purposes.
That's what calculators and computers do.
An infinite series is a formula with an infinite number of terms,
and you can only substitute in a finite number of them and get an
approximate answer.
For example,
x*sin(2/x) = .5 has an approximate solution of
x = 0.808219009492025...
and that would do for any practical application but it's not
an exact solution.
Edwin
