and i think the value is less than 1 right? certainly sin1 ≠ 1. here units i used are degrees.
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As other answer had said, θ must be in radian. This is because θ actually represent a ratio of length of an circle arc to the radius; while sine represents the ratio of half the chord to the radius. If you draw them on a picture, you will notice that these 2 lengths should be similar to each other: you are comparing the length of a chord to the arc length of the circle extending it.
When you do precise mathematics, saying “approximately” is no longer sufficient, you need to demonstrate how small the error is. Except for the case θ=0, sin(θ) is never equal to θ and there is always a small error, so you need to estimate it. As it turns out, the error is cubic in θ. That is, the absolute difference between θ and sin(θ) is never more than θ^3 . For example, if θ<=0.1, you can guarantee that the error is no more than 0.001, so you guarantee 3 significant figures.
How you we estimate this? It is a theorem from calculus called Taylor’s remainder theorem. In this case, it says that if you use θ to approximate sin(θ), then the error equals -θ^3 cos(t) for some t between 0 and θ, and even though we don’t know what exactly is t, we can guarantee that |cos(t)|<=1. The above estimate come from Taylor’s 3rd order approximation.
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