Finding Roots of b - tan(b) = 0

[rsaad]

How would you find the roots of:
b - tan (b) = 0

please do not that i have to plot the graphs of y=b and y=tan b and then i should find the solution. I want to know how to do it the other way.
thank you

 

[Mark44]

How would you find the roots of:
b - tan (b) = 0

please do not that i have to plot the graphs of y=b and y=tan b and then i should find the solution. I want to know how to do it the other way.

Another way, besides a graphical solution, is to use a numerical approximation technique such as Newton's Method (also known as Newton-Raphson). If you want to find out more, you can do a web search, which should generate lots of hits.

 

[lurflurf]

There are an infinite number. You can use an iterative method like Newton's mathod. There are also some asymptotic expansions, the large values of x are approximately pi/2+n pi for some large n.

 

[TheoMcCloskey]

For some iteration initial estimates, consider the following:

Let
$$c_m = \frac{2\,m + 1}{2} \, \pi$$

Then let the initial estimate $b^{(0)}$ be given by
$$b^{(0)} = c_m - u$$

Three possible initial estimates, in increasing accuracy, can be given by the three separate values of u as follows

$$u_2 = \frac{1}{c_m}$$
$$u_3 = \frac{1}{c_m} + \frac{1}{c_m^3}$$
$$u_5 = \frac{1}{c_m} + \frac{2}{3}\, \frac{1}{c_m^3}$$