Is there a way to solve the transcendental equation O=R{1-Cos(28.65S/R)} for R?

[damskippy]

Hey all,
I need some help rearranging for R.

O=R{1-Cos(28.65S/R)}

For those curious, its a Horizontal Sightline Offset formula used in geometric road design.

thanks

 

[jedishrfu]

Is that the letter O or zero to the left of the equals?

If zero then can't you just divide by R and solve what's left?

 

[Mentallic]

If it's not zero then that's the first time I've ever seen a variable being represented by the letter O. It would also mean that you can't explicitly solve for R.

 

[damskippy]

Apologies for the confusion, it's an O for Offset. This should make it a little clearer, (HSO being Horizontal Sightline Offset).

HSO=R{1-Cos(28.65S/R)}

But what your saying, you can't solve the above equation for R. Might explain why I couldn't, apart from my poor maths. Thanks for your help.

 

[Mentallic]

But what your saying, you can't solve the above equation for R. Might explain why I couldn't, apart from my poor maths. Thanks for your help.

Because we can't describe the solution to the above equation in terms of any finite number of algebraic operations (which include +,-,*,/,powers,roots).

For example, even though it's well known that the solution to e^x=2 is x=ln(2), this is a transcendental. We can't actually find the value of ln(2) exactly (althought it's irrational anyway) in terms of a finite number of algebraic operations. This function ln(x) was created and given a name because it's so commonly used.

The taylor series (which is an expansion of a transcendetal function with an infinite amount of algebraic operations to describe it) for the exponential is

$$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$$

So I believe if a computer were to approximate the value of e³ for example, it would truncate the taylor series for e^x and then plug in x=3 and solve that.

Similarly, your equation is transcendental, but the difference here is that there isn't any commonly known function to describe the solution. Surely someone could've named the solution whatever they liked, but it would still need to be solved numerically like the ln(x) function.