Question

Expand log sinx in powers of (x-2) by Taylor's Theorem.

Answer 1

Let $f(x)= \log \sin (x)$

$f(x)=f(2+(x-2))$

Now, as we know that $f(x)=f(a+h)$, where $a=2$ and $h=x-2$.

By Taylor's theorem we know that:

$f(x)=f(a)+h f'(a)+h^2\frac{f''(a)}{2!} +h^3\frac{f'''(a)}{3!} +...$

$\log \sin(x)=f(2)+(x-2)f'(2)+\frac{(x-2)^2}{2!}f''(2)+...$

$f(x)=\log \sin(x)$

$f'(x)=\frac{\cos x}{\sin x} = \cot x$

$f''(x)=-cosec^2x$

and so on.

Therefore, $f(2)=\log \sin(2)$

$f'(2)=\cot 2$

and so on.

Therefore,

$\log \sin(x)=\log \sin(2)+(x-2)\cot(2)+\frac{(x-2)^2}{2!} cosec^22+...$