Manual approximation of $\operatorname{sech}(0.7)$

Question

In the archive of a midterm exam collection there are some question like the one above.

How can we approximate expressions like $$\operatorname{sech}(0.7)$$ without a calculator?

Thanks in advance.

Answer 1

Note that $$cosh (x) = (e^x+e^{-x})/2 = 1+x^2/2 +x^4/24+..$$

Thus $$cosh (.7) \approx 1+(.7)^2/2+(.7)^4/24 =1.255$$

Therefore $$sech (0.7) \approx 0.7968$$

Answer 2

If you know that $\log 2 = 0.693\ldots \approx \frac{7}{10}$---this is certainly worth memorizing if you haven't---then you can deduce $$\operatorname{sech} \left(\frac{7}{10}\right) \approx \operatorname{sech} \log 2 = \frac{2}{\exp(\log 2) + \exp(-\log 2)} = \frac{2}{2 + \frac{1}{2}} = \frac{4}{5} ,$$ which has a relative error of $< 0.3\%$.