Find a cubic approximation for sin^-1 x

So I did part 1 and 2 (showing that from y=sin^-1 x that dy/dx = 1/(1-x^2)^1/2 and then expanding the first two terms of (1-x^2)^-1/2 to give 1 + 1/2x^2) but I do not understand what to do next on part 3 or 4.

It asks to find the cubic approximation for sin^-1 x but I do not understand what it wants. Part 4 asks by substituting x = (sqrt 3) / 2 into the result from part 3 find k if pi is approximately equal to k multiplied by sqrt3.

Reply 1

ghostwalker

17

Original post by aliman65

So I did part 1 and 2 (showing that from y=sin^-1 x that dy/dx = 1/(1-x^2)^1/2 and then expanding the first two terms of (1-x^2)^-1/2 to give 1 + 1/2x^2) but I do not understand what to do next on part 3 or 4.

It asks to find the cubic approximation for sin^-1 x but I do not understand what it wants. Part 4 asks by substituting x = (sqrt 3) / 2 into the result from part 3 find k if pi is approximately equal to k multiplied by sqrt3.

For part 3, they probably want you to integrate so you'd have sin^-1 as a truncated power series.

Reply 2

aliman65

OP

15

Original post by ghostwalker

For part 3, they probably want you to integrate so you'd have sin^-1 as a truncated power series.

The worksheet I am doing isn't based on integration and we have not covered integration yet.

Reply 3

ImpossibleGirl

You could integrate your answer from part 2, an approximation of dy/dx, to get an approximation of y maybe?

Reply 4

ghostwalker

17

Original post by aliman65

The worksheet I am doing isn't based on integration and we have not covered integration yet.

A somewhat poorer approximation would be to use:

\delta y = \frac{dy}{dx}\delta x

Basing it on y=0 when x=0 we have:

y = \frac{dy}{dx}x

Perhaps that's what they're looking for, but I'm really guessing now.

Reply 5

davros

16

Original post by aliman65

The worksheet I am doing isn't based on integration and we have not covered integration yet.

The standard way to do this is by integrating the series you've obtained.

I can't see what else would be expected here, unless you take a circuitous approach like this:

Suppose the required cubic expansion is

sin^{-1} x \approx a_0 + a_1x + a_2x^2 + a_3x^3

Differentiate this expression and equate coefficients with the series you already have!

Find a cubic approximation for sin^-1 x

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