X = n*cos(90-360/n)

Homework Statement

Given x=n*cos(90-360/n), express n in terms of x

Homework Equations

<no relevant equation since it's pure maths>

The Attempt at a Solution

All I can say is that the result is one of those functions, whose proper name I do not remember, with multiple n values corresponding to one x value. Not sure how to elegantly report that and this is mainly why I asked.

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haruspex
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Gold Member
2020 Award

Homework Statement

Given x=n*cos(90-360/n), express n in terms of x

Homework Equations

<no relevant equation since it's pure maths>

The Attempt at a Solution

All I can say is that the result is one of those functions, whose proper name I do not remember, with multiple n values corresponding to one x value. Not sure how to elegantly report that and this is mainly why I asked.
I don't see that there will be multiple n values for a given x. Looking at n=1, 2, 3.... x increases monotonically (and converges to a limit). On the other hand i would think it impossible to write n as a function of x in closed form.

I don't see that there will be multiple n values for a given x. Looking at n=1, 2, 3.... x increases monotonically (and converges to a limit). On the other hand i would think it impossible to write n as a function of x in closed form.

haruspex
Homework Helper
Gold Member
2020 Award
No, as I said, I see no way to get n as a function of x in closed form. Is this the complete statement of the problem?

Ray Vickson
Homework Helper
Dearly Missed

Homework Statement

Given x=n*cos(90-360/n), express n in terms of x

Homework Equations

<no relevant equation since it's pure maths>

The Attempt at a Solution

All I can say is that the result is one of those functions, whose proper name I do not remember, with multiple n values corresponding to one x value. Not sure how to elegantly report that and this is mainly why I asked.
I don't think the "result" involves one of those functions whose name you forget; it is true that the equation itself (NOT the solution) does involve the so-called "sinc" function. Using radians instead of degrees to represent angles, your equation is
$$x = n \,\cos \left(\frac{\pi}{2} - \frac{2 \pi}{n} \right).$$
Using ##\cos(\pi/2 - \theta) = \sin(\theta)##, this becomes
$$y = \frac{\sin(v)}{v},$$
where
$$y = \frac{x}{2 \pi}, \: \text{and} \; v = \frac{2 \pi}{n}$$

The function ##\sin(\theta)/ \theta## occurs frequently in applications, so has been given a name: ##\text{sinc}(\theta) = \sin( \theta)/\theta##. So, with ##y## and ##v## as defined above, your equation is ##\text{sinc}(v) = y##. You can see plots of the sinc funcion in http://mathworld.wolfram.com/SincFunction.html or
http://www.physics.usyd.edu.au/teach_res/mp/doc/math_sinc_function.pdf .

However, you should realize that there are no known formulas for the solution of that equation, so you cannot hope to express n in terms of x as any kind of known function. All you can do is solve it numerically for various values of x and maybe plot or tabulate the numerical results.