[转载]为什么当一个角theta（θ）很小时sin theta约等于theta

Posted 2021-08-18 无声烟雨

以下内容转载自https://math.stackexchange.com/questions/870092/why-is-sind-phi-d-phi-where-d-phi-is-very-small

其中一个答案是：

Just draw the diagram!

What does sinx mean? it\'s the ratio of the opposite side to the hypotenuse in a triangle.

Now, let\'s draw a triangle with a small angle x inside the unit circle:

Now clearly, when the angle becomes really small, the opposite side is approximately the arc length. In radians, the arc length in a unit circle is exactly the angle x, and so we have for small angles:

另一个答案是：

If you are familiar with Taylor series you know that the series of sin(x) expanded at 0 is:

Then, if x is very small you can neglect all term of order greater than one getting:

You can also show this result using basic trigonometry but this approach seems easier to me.

还有一个答案是：

You can give a linear approximation for sin near 0 based on this formula:

and using the fact that: sin′=cos, you get:

So when x is very small, you have that sinx∼x.

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What this intuitively means, is that when you observe closely the graph of the curve sinx near 0, it starts to resemble a line, and this line is described by y=x.

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