Maclaurin's Series Question

[kaizerlia]

3. Given that \(\displaystyle \, y\, =\, \sin^{-1}(x),\,\) prove that \(\displaystyle \, (1\, -\, x^2)\,\dfrac{d^2y}{dx^2}\, -\, x\,\dfrac{dy}{dx}\, =\, 0.\,\) By further differentiation of this result, find the series expansion of y in ascending powers of x up to and including the term in x³. Hence,

(i) by choosing a suitable value of x, express the value of \(\displaystyle \, \pi\,\) in the form \(\displaystyle \, \dfrac{p}{q},\,\) where p and q are integers to be determined, and

(ii) explain, with the aid of graphs, why x = 1 is not suitable choice for (i).

I've already gotten the answer for the Maclaurin expansion, which is y= x + 1/6(x^3). Unless I'm wrong about that too

For Part (ii) I've drawn to the sin curve and referred to it in my answer to explain why x=1 is not a suitable choice.
However, I'm not sure as to what approach I should take to answer Part (i) and it's really frustrating because I have the feeling that it's actually a very simple solution... I've equated the given equation to the Maclaurin expansion--
sin⁻¹ (x) = x + 1/6(x^3). But I do not know how to proceed from there onwards? The only way I can think of to get pi from the equation is if I substitute x=0 into the solution.. but that will not give me an answer remotely close to pi.

I've found myself stuck on similar kinds of questions as well. For example, questions like using a suitable substitution, find an approximation to sq/cube root (value)... is there any recommended approach I can adopt to tackle these questions better?

Thank you

 

[Ishuda]

3. Given that \(\displaystyle \, y\, =\, \sin^{-1}(x),\,\) prove that \(\displaystyle \, (1\, -\, x^2)\,\dfrac{d^2y}{dx^2}\, -\, x\,\dfrac{dy}{dx}\, =\, 0.\,\) By further differentiation of this result, find the series expansion of y in ascending powers of x up to and including the term in x³. Hence,

(i) by choosing a suitable value of x, express the value of \(\displaystyle \, \pi\,\) in the form \(\displaystyle \, \dfrac{p}{q},\,\) where p and q are integers to be determined, and

(ii) explain, with the aid of graphs, why x = 1 is not suitable choice for (i).

I've already gotten the answer for the Maclaurin expansion, which is y= x + 1/6(x^3). Unless I'm wrong about that too

For Part (ii) I've drawn to the sin curve and referred to it in my answer to explain why x=1 is not a suitable choice.
However, I'm not sure as to what approach I should take to answer Part (i) and it's really frustrating because I have the feeling that it's actually a very simple solution... I've equated the given equation to the Maclaurin expansion--
sin⁻¹ (x) = x + 1/6(x^3). But I do not know how to proceed from there onwards? The only way I can think of to get pi from the equation is if I substitute x=0 into the solution.. but that will not give me an answer remotely close to pi.

I've found myself stuck on similar kinds of questions as well. For example, questions like using a suitable substitution, find an approximation to sq/cube root (value)... is there any recommended approach I can adopt to tackle these questions better?

Thank you

What is, for example, sin(\(\displaystyle \frac{\pi}{2}\)) and thus what would y be when x were that number? What other values of the sine sould you use to get a better answer