Solving a Simple Equation for Gamma: Tips and Tricks

[Norman]

I am trying to solve this equation:

Tan[\gamma a]=-\frac{\gamma}{\beta}

where \beta and a are just numbers and I am trying to solve for \gamma. I tried graphing it but I don't see how the solution varies with the choice of a, which is a free parameter in the problem.

 

[Cummings]

You can solve by drawing the graph of both Tan[xa] and -x/b and seeing when they intersect. (i used x in place of the symbol you gave)

thus it has many solutions as tan is a trigonometric function.

If i took a as 4 and b as 5 then
x can equal 0, Tan[0] = 0
x can equal 3, Tan[12] = -3/5 (approximatly)

this is from using a graphics calculator and finding where the two graphs intersect.

There are infinate more values.

i do not know of any other way to solve for the unknown. Yet.

 

[Norman]

Yes I understand how to do that... maybe I didn't phrase my question well. By holding \beta constant and varying a, I can obtain and bunch of numbers for \gamma and then fit the curve to obtain how \gamma varies. What I was wondering was if there was another way to do this, that doesn't force me to use curve fitting. Any ideas anyone? Or maybe an easy way to see how \gamma varies when changing a.
Thanks.

 

[cookiemonster]

\tan {\gamma a} = -\frac{\gamma}{\beta}

Treat a as a function of gamma and differentiate.

\sec^2{\gamma a} \Big( a + \gamma \frac{da}{d\gamma} \Big) = -\frac{1}{\beta}

Solve for d\gamma.

d\gamma = \frac{-a \beta da}{\cos^2{\gamma a} + a\beta}

cookiemonster