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