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I am just trying to understand sine.

I know how it works, but I don't know why it works that way.

What exactly is result of $\frac{opposite(height)}{hypotenuse (radius)}$?

What is behind this magic operation?

I know I can use it to find out an angle of triangle in sinusoide.

I've read a lot of topics and I've seen a lot of videos about this, but I still don't get it.

Thanks a lot!

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1 Answer

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It's not exactly clear what you mean by "why it works" that way. The sine of some angle $\theta$ is defined as the ratio of the lengths of two sides of some arbitrary right triangle drawn about $\theta$ - specifically, we are taking the ratio of the lengths of the "opposite" side to the hypotenuse. It is simply a number that represents the ratio of the two side lengths.

Perhaps you are wondering why the sine of some fixed angle is always the same, despite the triangle, or why the sine of an angle on the unit circle is a sufficient representation of the sine all such angles... This works because all triangles with the same three angle measurements are similar. In other words, any triangle with the same three angles will have sides whose lengths are a constant multiple of any other triangle with the same three angles; hence, the ratios of the lengths between any two arbitrary sides will always remain the same between triangles.

Imagine we have an angle $\theta_1$. If we draw a right triangle about the angle $\theta_1$, then the third remaining angle is $\theta_2 = 180 - 90 - \theta_1$. Now, any right triangle drawn about the angle $\theta_1$, no matter what the side lengths are, will have the same three angle measurements and thus be similar. Therefore, the sine of the angle $\theta_1$ is invariant.

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