Typesetting multi-line equations

[DrClaude]

I have a question on how to properly typeset a series of inequalities or approximate equalities when the LHS does not change. Take for example
$$f(x) = \sin(x) \\ \quad \approx x - \frac{x^3}{3!} \\ \quad = x - \frac{x^3}{6}$$
What I did there is that I took it as if it was one long line,
$$f(x) = \sin(x) \approx x - \frac{x^3}{3!} = x - \frac{x^3}{6}$$
that is split and stacked. Is this the correct way to do it? Or is it assumed that the LHS repeats, i.e.,
$$f(x) = \sin(x) \\ f(x) \approx x - \frac{x^3}{3!} \\ f(x) = x - \frac{x^3}{6}$$
in which case the last line is incorrect?

 

[jedishrfu]

I would for clarity use the aproximately equals in the last equation and repeat the LHS.

Why?

Because as you're composing your solution steps you might need to insert a step and by not explicitly typing the LHS for each line means confusion will set in especially if you're reviewing your work weeks or months later for a test.

 

[DrClaude]

I would for clarity use the aproximately equals in the last equation and repeat the LHS.

What I don't like about your approach is that once you introduce a $\approx$, there are no more $=$, so that it is not obvious if additional approximations are made.

There is also a problem when the LHS is itself very long.

 

[AlephZero]

My basic "rules" would be
(1) never write anything that is actually wrong.
(2) assume the reader at least knows enough to follow the mathematical argument.

So in the OP, all the options are OK, except for the last line $$f(x) = x - \frac{x^3}{6}$$ which just plain wrong.

In a more complicated situation you might need to spell out a detail like "and since 3! = 6 we get" ...

 

[DrClaude]

(2) assume the reader at least knows enough to follow the mathematical argument.

I like that