In trigonometry, how can we find the sine/cosine of an angle larger than 90 degrees, if sine and cosine only work for right triangles?

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In trigonometry, how can there be angle thetas larger than 90 degrees, if sine and cosine only work for right triangles? I realize what we do is that we take the sine or cosine of the angle complementary to it. But then, how does that give the ratio between the opposite/adjacent and hypotenuse of the angle larger than 90 degrees? It doesn't! It only gives the ratio between the opposite/adjacent and hypotenuse of the angle complementary to it! Can someone please explain?