SOH

SOH is a mnemonic used in trigonometry to remember the relationship between the angles and sides of a right triangle. It stands for Sine equals Opposite over Hypotenuse. Together with CAH TOA, it forms a common teaching aid to recall the three primary trigonometric ratios: sine, cosine, and tangent.

In a right triangle, for an acute angle theta, the standard definitions are: sin(theta) = opposite/hypotenuse; cos(theta) = adjacent/hypotenuse; tan(theta) = opposite/adjacent. These ratios depend only on the shape of the triangle and not on its size, making them useful for solving problems involving similar triangles.

Applications of SOH CAH TOA include calculating an unknown side when a side length and an angle are known, or determining an angle when two sides are known. The sine and cosine values are dimensionless and, for a given angle, are confined to the range between -1 and 1. In the context of a right triangle with an acute angle, all three ratios yield positive values.

Limitations should be noted: the SOH CAH TOA mnemonic applies specifically to right triangles. For non-right triangles, trigonometric solutions use other principles such as the laws of sines and cosines. In broader mathematical practice, sine and cosine are defined for all angles via the unit circle, extending their use beyond right triangles, but SOH CAH TOA remains a helpful introductory memory aid for basic right-triangle work.