New user, need help with a trigonometry equation for physics Tan x = A + B sin x

Hi, I am a physics teacher and am working on a problem for a general solution to the problem of the swing carousel ride. Most physics students learn the to solve the problem of finding the angle of the swings given the tangential speed , or vice versa. This is easy.

A harder task is to find the angle as a function of angular velocity, and radius of the platform from which hang the swings. Looking only for the steady state solution, I decided I would have to work at it from the non-inertial rotation frame of reference, and use the pseudoforce centrifugal force. It make it easier to get an equation for the angle that only depends on the constant parameters of angular speed , platform radius , chain length , and gravity g.

the problem is I don't know how to solve for the angle. I could get a numerical solution but that is not what I want.

The equation boils down to

Tan x = A + B sin x

where A and B are constants that depend on the parameters. What should I do to solve for x?

Thanks for any help!

Eric

Re: New user, need help with a trigonometry equation for physics Tan x = A + B sin x

Hi Eric!

You can rewrite tan x = A + B sin x as:$\displaystyle {\sin x \over \pm \sqrt{1-\sin^2 x}} = A + B \sin x$

Substitute y=sin x, and you get:$\displaystyle {y \over \pm \sqrt{1-y^2}} = A + B y$

which yields:$\displaystyle y^2 = (A + B y)^2 (1 - y^2)$

This is a 4th order polynomial, which can be solved as explained for instance here: Quartic function - Wikipedia, the free encyclopedia

This may be more involved than you'd like, although I have to say that the article makes it appear more difficult than it actually is.

Afterward, you get the answer with $\displaystyle x = \arcsin y$.

Is this what you had in mind, or were you hoping for an easier solution?

Re: New user, need help with a trigonometry equation for physics Tan x = A + B sin x

post trig problems in the trig forum, please. the lobby is for introductions only.

Re: New user, need help with a trigonometry equation for physics Tan x = A + B sin x

Thanks very much for the idea. I should have thought of that with the tan function but I would have had to ask anyway for help with the next step. I will have a look at the Quartic function - Wikipedia, the free encyclopedia suggestion.

Cheers,

Eric

Re: New user, need help with a trigonometry equation for physics Tan x = A + B sin x

You're welcome!

Till next time (probably in a different sub forum ;-)).