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.

Figure 1: The corners of a triangle are labelled with capital letters A, B, and C. The opposite sides of these corners are labelled with the lower case letters a, b, and c. This allows for the sine law to describe the sides and corners of a triangle using these letters to identify them.

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 (Eq. 1), or vice versa (Eq. 2).

Eq. 1: \({a \over {\sin A}}={b \over {\sin B}}={c \over {\sin C}}\)

Eq. 2: \({{\sin A} \over a}={{\sin B} \over b}={{\sin C} \over c}\)

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).

Figure 2: A triangle which has a known length of a side and a known angle of an opposite corner can be solved using the sine law. The sine law dictates that the ratio of the sine of an angle and the length of its opposite corner are the same for all corner-angle combinations in the triangle. After calculating this ratio for the known side-corner combination, this ratio can be used along with the known side length to calculate the angle of unknown opposite corner, or along with the angle of a known corner to calculate the length of an unknown side.

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) (Eq. 3), which we can subsequently rearrange into an equation to find the angle of the corner (B) (Eq. 4), giving a length of 55cm.

Eq. 3: \({{\sin 39.4} \over 35}={{\sin 85.9} \over b}\)

Eq. 4: \({{\sin 85.9} \over {{\sin 39.4} \over 35}}=b=55\)

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 (Eq. 5). 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 (Eq. 6).

Eq. 5: \({{\sin 39.4} \over 35}={{\sin C} \over 45}\)

Eq. 6: \(\sin^{-1}({{\sin 39.4} \over 35}) \times 45=C=54.7\)