SOHCAHTOA

SOHCAHTOA is a pneumonic device used to remind students of the three trigonometric functions and their equivalents on a right triangle. Problems utilizing SOHCAHTOA will often have students solve for a missing side length on a triangle, or occasionally find a formula to express a side length (where there may only be variables or a mixture of numbers and variables, but where a final number cannot be solved for).

One tip for remembering the letters themselves is that each of the side labels (adjacent, opposite, and hypotenuse) is used twice. There are two Os, two As, and two Hs. Remember either that Tan doesn’t use hypotenuse or that Sin and Cos do, and the rest of the relationships will fall into order.

It’s also worth remembering that these identities are true for both acute angles of a right triangle. And since the labels of “adjacent” and “opposite” are relative to which side is being used, there is some overlap between these identities.

Sin and Cos Overlap

The Sine of the 1st acute angle of a triangle is equal to the Cos of the other acute angle, and vice versa. In our examples above, sinA = cosB and sinB = cosA.

Word Problems

SOHCAHTOA is most common on problems that look like our image above, where there is a triangle on its own and the problem will look to have students solve for a given angle or side. But SOHCAHTOA can also be used to solve word problems.

The triangle may already be drawn in (like the example below) or the question may simply describe a situation that creates a right triangle. The key thing to remember is that: right triangle + trig functions = SOHCAHTOA. 

The example below is from the July 2019 ACT. The problem references the following information provided above the question itself: the maximum altitude during Tour C is 1,000 feet.

The angle 37 is used in all of the answer choices, and so the answer will involve SOHCAHTOA, the 37-degree angle, and the height of 1,000 feet.

The question asks about the length of the hypotenuse of the triangle and the question gives the length of the side opposite the angle. Therefore, this is a situation where SOH (

sin = OppositeHypotenuse)will be relevant.

sin37 = 1,000c A little algebra then leads to: 

c =1,000sin37. F is our answer.