Calculators either use the Taylor Series for $\sin / \cos$ or the CORDIC algorithm. A lot of information is available on Taylor Series, so I'll explain CORDIC instead.
The input required is a number in radians $\theta$, which is between $-\pi / 2$ and $\pi / 2$ (from this, we can get all of the other angles).
First, we must create a table of $\arctan 2^{-k}$ for $k=0,1,2,\ldots, N-1$. This is usually precomputed using the Taylor Series and then included with the calculator. Let $t_i = \arctan 2^{-i}$.
Consider the point in the plane $(1, 0)$. Draw the unit circle. Now if we can somehow get the point to make an angle $\theta$ with the $x$-axis, then the $x$ coordinate is the $\cos \theta$ and the $y$-coordinate is the $\sin \theta$.
Now we need to somehow get the point to have angle $\theta$. Let's do that now.
Consider three sequences $\{ x_i, y_i, z_i \}$. $z_i$ will tell us which way to rotate the point (counter-clockwise or clockwise). $x_i$ and $y_i$ are the coordinates of the point after the $i$th rotation.
Let $z_0 = \theta$, $x_0 = 1/A_{40} \approx 0.607252935008881 $, $y_0 = 0$. $A_{40}$ is a constant, and we use $40$ because we have $40$ iterations, which will give us $10$ decimal digits of accuracy. This constant is also precomputed1.
Now let:
$$ z_{i+1} = z_i - d_i t_i $$
$$ x_{i+1} = x_i - y_i d_i 2^{-i} $$
$$ y_i = y_i + x_i d_i 2^{-i} $$
$$ d_i = \text{1 if } z_i \ge 0 \text{ and -1 otherwise}$$
From this, it can be shown that $x_N$ and $y_N$ eventually become $\cos \theta$ and $\sin \theta$, respectively.
1: $A_N = \displaystyle\prod_{i=0}^{N-1} \sqrt{1+2^{-2i}}$
When $\sin$ is defined geometrically, it is typically as the vertical coordinate on the unit circle. Certainly this is the only sensible extension from $[0^\circ, 90^\circ]$ if you want to retain any of the nice properties like analyticity, angle addition formulas, Euler's formula, etc.
Best Answer
One could think as follows: Since the side opposite to the angle B is the hypotenuse, then: $$ sinB = \frac{hypotenuse}{hypotenuse} = 1$$
However, one must always look carefully and rigorously to definitions. The sine of an angle is defined as the division of the opposite side to the angle by the hypotenuse. The hypotenuse can't be one of the sides. Having a look at the trigonometric circle might help:
The sine is defined here always as the value of $y$ divided by 1 (hypotenuse) When the angle is 90°, then $y=1$. I don't want to sound rude, but there is not a lot of sense in "choosing the sides" for calculating the sine of 90°. Hope i've helped.