# How would you solve something like asint=bt or acost=bt for t?

## For example, $8 \cos x = 3 x$ $5 \sin y = 5 y$

Apr 10, 2017

Such equations generally have no algebraic solution, so you need to use a numerical method.

#### Explanation:

As Steve said, you would solve such equations using a numerical method such as Newton-Raphson:

Given a differentiable function $f \left(x\right)$ for which we want to find a zero, start with a first approximation ${a}_{0}$, then repeatedly apply the formula:

${a}_{i + 1} = {a}_{i} - \frac{f \left({a}_{i}\right)}{f ' \left({a}_{i}\right)}$

For example:

To find approximations to $x$ satisfying $8 \cos \left(x\right) = 3 x$, define:

$f \left(x\right) = 8 \cos \left(x\right) - 3 x$

Then:

$f ' \left(x\right) = - 8 \sin \left(x\right) - 3$

Put the following formula into a spreadsheet:

${a}_{i + 1} = {a}_{i} + \frac{8 \cos \left(x\right) - 3 x}{8 \sin \left(x\right) + 3}$

Then we find:

${a}_{0} = 0$

${a}_{1} = 2. \overline{6}$

${a}_{2} \approx 0.3965853162202127$

${a}_{3} \approx 1.4128671965703734$

${a}_{4} \approx 1.1394457744448268$

${a}_{5} \approx 1.1322818941294521$

${a}_{6} \approx 1.1322734714493981$

${a}_{7} \approx 1.1322734714376355$

${a}_{8} \approx 1.1322734714376355$

So it converges in a few steps to a good approximation of a (in fact the) solution.

If there is more than one solution, then you might try different values for ${a}_{0}$.