Approximating sine for small values

Recall that

$$\lim_{x\rightarrow 0} \frac{\sin(x)}{x} = 1$$

One way to interpret this statement is that for |x|<<1 we have

$$\frac{\sin(x)}{x}\approx 1$$

or

$$\sin(x) \approx x$$

or

$$\sin(x) -x \approx 0.$$

How like is $\sin(x) -x$? Well, we could compare it to a power of x. Since $\sin(x) -x$ is odd, it would make sense to compare it to x³ and look at the behaviour of

$$\frac{\sin(x) - x}{x^3}$$

as x tends to . L'Hopital's rule can be tried here. The ratio of the derivatives is

$$\frac{\cos(x) - 1}{3x^2} = \frac{-1}{3(\cos(x) + 1)}\left(\frac{\sin(x)}{x}\right)^2$$

which clearly has a limit of -1/6 as x approaches . Therefore

$$\lim_{x\rightarrow 0}\frac{\sin(x) - x}{x^3} = -\frac{1}{6}.$$

Interpreted as an approximation we see that for |x|<<1 that

$$\sin(x) \approx x - \frac{x^3}{6}$$

Here is a picture indicating how good the approximation is. We graph $\sin(x) - (x - x^3/6)$ from to $\pi/8$ and from $\pi/8$ to $\pi/4$.

Note that we don't do to well in the latter interval, so we may want to improve our approximation by repeating the procedure again. $\sin(x) - (x - x^3/6)$ is

Once again, $\sin(x) - (x - x^3/6)$ is again odd and approximately when |x|<<1 so this time we try x⁵ and look at

$$\frac{\sin(x) - x + x^3/6}{x^5}$$

as x tends to . L'Hopital's rule can be tried here. The ratio of the derivatives is

$$\frac{\cos(x) - 1 + x^2/2}{5x^4}.$$

It is not clear what the limit is here, but L'Hopital's rule may be tried for this new ratio giving the ratio

$$\frac{-\sin(x) + x}{20x^3} = \frac{-1}{20}\frac{\sin(x) - x}{x^3}$$

and we recognize that the limit here is -1/120 as x approaches . Applying L'Hopital's Rule twice, we see that

$$\lim_{x\rightarrow 0}\frac{\sin(x) - x + x^3/6}{x^5} = \frac{-1}{120}$$

Interpreted as an approximation we see that for |x|<<1 that

$$\sin(x) \approx x - \frac{x^3}{6} + \frac{1}{120}x^5.$$

Here is a picture indicating how good the approximation is. We graph $\sin(x) - (x - x^3/6 + x^5/120)$ from to $\pi/8$ and from $\pi/8$ to $\pi/4$. Note that we do much better in the latter interval compared with the last time.