Using $\ln (\cos x)=\frac{-x^2}{2}-\frac{x^4}{12}+...$, approximate $\ln 2$ in terms of $\pi$

Question

Using $f(x)=\ln (\cos x)=\dfrac{-x^2}{2}-\dfrac{x^4}{12}+\dots $, approximate $\ln 2$ in terms of $\pi$.

I know $\cos(x)$ will never be two - so what can I actually substitute in to get something in terms of pi?

Answer 1

Notice that the natural logarithm has a useful property $$\ln\left(a^b\right)=b\ln(a)$$ So in fact, we only have to get a value of cosine that is an exponent of $2$ to approximate $\ln(2)$.

Further notice that $$\cos\left(\frac \pi 4\right)=\frac {1}{\sqrt{2}}$$