>"But I thought sine was the ratio of the opposite side to the hypotenuse of a right-angled triangle? What is sine of 120 degrees? What does that mean?".
Yes, sine is the ratio of the opposite side to the hypotenuse. We use a *unit circle* so that the hypotenuse is always one. This way, the "opposite side" always stays inside the unit circle, so it ranges .
The unit circle is one unit on purpose. We don't need to write the denominator when it's 1. (Every number has an invisible denominator of 1.) For every point along the circle, the sine and cosine values are linked by a2 + b2 =1. If we know sine, we can find cosine and vice-versa. Both of them have invisible denominators of 1. Both of them are bound within the unit circle with values ranging from -1 to +1.
For example, the sine of 30 degrees is 1/2. That same sine value could be seen on a triangle with a vertical side of 1 and a hypotenuse of 2 (horizontal side is square root of 3). The same sine value, 1/2 could also be seen inside a unit circle with a vertical side of 1/2 and a hypotenuse of 1 (horizontal side of square root of 3 over 2). It's the same triangle, just scaled to fit the unit circle so we can all agree on the values as a function of the hypotenuse.
In the calculator, the sine and cosine are the unit circle values, the ratios with a denominator of 1. Then tangent is the ratio of sine to cosine. And every other trig function is a form of sine, cosine or combination of both.