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Sine of Angle and Angle in Radians
As stated here, the angle θ in radians is defined by considering the arc of a circle

A right angled triangle including θ has been added to this diagram of a segment of a circle.

From this, we have

It should be clear from this diagram that these two ratios are not equal (s is noticeably longer than y).
For example, if θ = 60° then

Here the angle is smaller, maybe about 30°.
Now we find

Now consider an even smaller angle (diagram enlarged to be able to put the letters on it!)

It is now clear that y is nearly the same length as s.

If this angle is 10° then we now have

So, even at 10°, the angle is now small enough for these two ratios to be equal, to three decimal places.

N.B.
Remembering that tanθ is defined as

we could use similar reasoning to find that for small angles tanθ is also equal to θ in radians.

In conclusion, for small angles, (usually considered to mean 3° or less):

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