find R???

Let's go from  Melody's last step...

4.543 = R (1 + .7049/R)      ....distribute R through on the right

4.543 = R + .7049             .......subtract .7049 from both sides and that will give you "R'

$$4.543=R\left(1-\frac{cos(28.65*200.53)}{R}\right)\\\\ 4.543= R\left(1-\frac{cos(5745.1845)}{R}\right)\\\\ 4.543\approx R\left(1-\frac{-0.7049}{R}\right)\\\\ 4.543\approx R\left(1+\frac{0.7049}{R}\right)\\\\$$

Now multiply both sides by R, take it all to one side and put =0, solve it as a quadratic equation.

I have not done this so I don't know what the answer will be.

You do it - if you have problems then post again. 

Thanks Chris,

Silly me, you did not need to multiply both sides by r you just needed to simplify the right side and solve.

It is simple.  Shame on me!