While most trigonometry problems can be solved by constructing right triangles, some special trigonometry laws can also be applied to directly solve oblique triangles when certain data are known. Three of these special laws for oblique triangles are the law of sines , the law of cosines , and the cotangent law . In many applications, much calculation can be avoided by using these laws, and the solutions to the problems are often quicker and easier to handle.

The Law of Sines

If any triangle—right, acute, or obtuse—is examined, it is readily apparent that the longest side always lies opposite the largest angle and the shortest side always lies opposite the smallest angle. This relationship leads to the law of sines, which states that in any triangle, the ratio of the sine of an angle to its opposite side is equal to the same ratio for another angle and its opposite side.

Consider Figure 21.12 in which acute oblique triangle ABC is shown. If altitude h is drawn from angle B to the opposite side b , then by definition of the sine function: