logo

For an Angle Small Enough 📂Mathematical Physics

For an Angle Small Enough

Description

Physics makes use of the approximation sinxx\sin x\approx x in many places. The reason this approximation can be used is because the following equation holds:

limx0sinxx=1 \lim \limits_{x\rightarrow 0}\frac{\sin x}{x}=1

Since this equation is first introduced in high school, college students might feel it is obvious enough to not question the validity of such approximation. However, how small should something be to be considered similar? For instance, when calculating the period of a simple pendulum using this approximation, one might wonder ‘How much should the pendulum oscillate for the approximation to hold?’ Let’s look at the graphs of y=sinxy=\sin x and y=xy=x.

untitled1.png

Looking at the graph, it is clear that as x\left| x \right| becomes smaller, the two graphs become almost identical. If we mark the point where the differences start to become noticeable,

untitled.png and calculate the ratio, it turns out to be 55395871=0.943450\dfrac{5539}{5871}=0.943450. We can see that an error of about 66% occurs. This seems close enough to be considered similar. Now, let’s plot the graph of y=sinxxy=\dfrac{\sin x}{x}.

3.png

The angle at which an error of about 55% occurs is in radians, 0.05360.0536. The angle at which an error close to 1010% occurs is in radians, 0.77750.7775. Converting this to degrees {}^{\circ},

2020-02-0715;45;02\_.png

One can see that it’s quite a generous range. Just because an angle is considered “sufficiently small” does not necessarily limit it to within 1010^{\circ} or 55^{\circ}. It might depend on how much error is tolerable, but within 3030^{\circ} the error between sinx\sin x and xx is less than 55%, so it seems reasonable to consider angles smaller than 3030^{\circ} to be “sufficiently small”.