SOHCAHTOA is a mnemonic device for the sine = (opposite side)/(hypotenuse) or SOH, cosine = (adjacent side)/(hypotenuse) or CAH, and tangent = (opposite side)/(adjacent side) or TOA . These represent the ratios of any of the two acute angles of a right triangle.

SOHCAHTOA facilitates recall on what ratio is needed for calculation based on the given problem. Here are some examples of how to use these ratios:

• When the opposite (O) and adjacent (A) side of an acute angle is given, recall TOA, and then solve for the tangent ratio. One can use the SHIFT + tan key combination (to activate the inverse tangent function) of the calculator to solve for the acute angle whose opposite and and adjacent sides are given.
• When the opposite (O) side and the hypotenuse (H) are given, we can use SOH to help us recall that we can now solve for the sine of the acute angle. Use the SHIFT + sin key combination (to activate the inverse sine function) of the calculator to solve for this acute angle.
• When an acute angle θθ is given and the hypotenuse (H), we recall CAH, and then solve for the adjacent side (A) using the equation A = H cos(θ).cos?(θ). We recall SOH, and then solve for the opposite (O) side using the equation O = H sin(θ).sin?(θ).

Note:

• Verify that the appropriate mode is chosen for the measurement of the angle. Oftentimes, we use the DEG or degree mode for solving triangles. However, if the angle measurements are expressed in terms of of π,π, it is sometimes worth using the RAD or radians mode rather than first converting the angle measure to degrees then using the DEG mode.

• Since the non-right angles of a right triangle are acute, using any of the key combinations for activating the inverse trigonometric functions: SHIFT + sin, SHIFT + cos, and SHIFT + tan, should give you the angle measure of the acute angle of interest. (Note: For solving obtuse triangles, sometimes we may need to add 180? or π,180? or π, as the situation may be, when we use SHIFT + tan since the tangent ratio may be negative and some calculators give as inverse tangent a negative angle in QIV, an angle that is measured clockwise instead of counterclockwise from the positive x-axis. A similar problem arises when we need to find the inverse sine of a negative value by using the SHIFT + sin key combination. In this case, to get the the angle in standard position that has the smallest positive measure and that has the given sine value, add 180° or π, as the case may be, to the absolute value of the the negative angle given by the calculator.')

SOHCAHTOA and the methods mentioned above will be oftentimes used to completely solve a triangle in the following cases:

• Case 1: the lengths of two sides; or,
• Case 2: the length of a side and the measurement of an acute angle.

(Case 1) To use SOHCAHTOA on a scientific calculator for solving triangles when two sides are given:

• Using the Pythagorean Theorem, solve for the missing side.
• Now that the three sides are known, obtain the ratios sine, cosine, and tangent of the acute angle of interest of the given right triangle.
• Verify that the appropriate mode is chosen for the measurement of the angle. Oftentimes, we use the DEG or degree mode for solving triangles. However, if the angle measurements are expressed in terms of of π,π, it sometimes worth using the RAD or radians mode to rather than convert the angle measure to degrees then using the DEG mode.

• Since the non-right angles of a right triangle are acute, using any of the key combinations for activating the inverse trigonometric functions: SHIFT + sin, SHIFT + cos, and SHIFT + tan, should give you the angle measure of the acute angle of interest. (Note: For solving obtuse triangles, sometime we need to add 180? or π,180? or π, as the situation may be, when we use SHIFT + tan, since the tangent ratio may be negative and some calculators give negative angle measures in QIV as answers to SHIFT + tan when the tangent ratio is negative. Adding π gives an angle in QII.)

(Case 2) Suppose the length of a side and a measurement of one of the acute angle are given.

• First, use one of the trigonometric ratios to solve for one other side, then proceed as in Case 1.
• Example: If an angle and its opposite (O) side are given, recall SOH, and use the sine ratio to solve for the hypotenuse. This opposite side of the given acute angle is the adjacent side of the other acute angle. Now knowing the hypotenuse and an adjacent side (recall CAH) of this second acute angle, we can use the cosine ratio and the inverse cosine function (SHIFT + cos) to solve for the measure of this second acute angle. We still need to solve the third side: we can either use the Pythagorean Theorem or TOA for one of the acute angles to find the missing side.