The law of sines is a law that describes how the sides and angles in a triangle relate to one another (Figure 1). This law states that for any triangle, there is a consistent ratio between the sine of an angle at a corner, and the length of the opposite side, that is the same for any corner:side combination in the same triangle.
The sine law can be written as an equation, and works whether the length of the sides are divided by the sines of the angles of the corners, or vice versa.
$${a \over {\sin A}}={b \over {\sin B}}={c \over {\sin C}}$$
$${{\sin A} \over a}={{\sin B} \over b}={{\sin C} \over c}$$
If the maximum precision is used, the sine law applies to any triangle, however there will likely be some discrepancies due to imprecision in measuring either the length of sides or the angle of corners. If this is an issue, consider the accuracy of measurement required for your triangle measurement.
Using The Sine Law
Where we have a known length of a side and an angle of an opposite corner, we can use the sine law to find the lengths of unknown sides and the angles of unknown corners (Figure 2).
If we have a known angle of a corner (A) and length of side (a), we have a sine law ratio. We can put these numbers into an equation along with the length of a known side (b), which we can subsequently rearrange into an equation to find the angle of the corner (B), giving a length of 55cm.
$${{\sin 39.4} \over 35}={{\sin 85.9} \over b}$$
$${{\sin 85.9} \over {{\sin 39.4} \over 35}}=b=55$$
Similarly if you want to find the angle of a missing corner, you can rearrange the same equation to be for the angle of the corner (C), by populating the equation with the known side-angle combination, along with the length of side c. Rearranging this equation, and using the inverse sin function sin^-1 gives the final equation for using the sine law to find the angle of a corner.
$${{\sin 39.4} \over 35}={{\sin C} \over 45}$$
$$\sin^{-1}({{\sin 39.4} \over 35}) \times 45=C=54.7$$