SOH-CAH-TOA Calculator

SOH-CAH-TOA is a memory aid for the three primary right-triangle ratios. SOH means sine equals opposite over hypotenuse. CAH means cosine equals adjacent over hypotenuse. TOA means tangent equals opposite over adjacent. The labels opposite and adjacent are always relative to the acute angle you selected.

Because the definitions "opposite" and "adjacent" assume a fixed reference angle paired with a 90-degree angle, which together force the third side to be the hypotenuse. In a non-right triangle there is no hypotenuse and no canonical right-angle vertex, so you cannot label the sides as opposite or adjacent in a way that makes sin = opp/hyp meaningful. For oblique triangles (no 90-degree angle), trigonometry uses the law of sines (a/sin A = b/sin B = c/sin C) and the law of cosines (c^2 = a^2 + b^2 - 2ab cos C) instead. Both reduce to SOH-CAH-TOA and the Pythagorean theorem when one angle is exactly 90 degrees, so the right-triangle case is just the special, simpler version.

They are read relative to whichever acute angle you call the reference angle, never the right angle. The opposite side is the leg that does not touch the reference angle (it sits across the triangle from it). The adjacent side is the leg that does touch the reference angle and is not the hypotenuse. The hypotenuse is always the side opposite the right angle and is always the longest side. Swap which acute angle is the reference and the opposite and adjacent legs swap with it, so sin(A) = cos(B) for the two acute angles A and B (they are complementary, A + B = 90 degrees, which is where the name cosine, meaning "complement's sine", comes from).

They are the inverse trig functions: arcsin (also written sin^-1) takes a ratio and returns the angle whose sine equals that ratio. Same for arccos and arctan. You need them whenever you know two sides of a right triangle and want the angle, which is exactly the "solve for angle" mode in this calculator: if opposite = 3 and hypotenuse = 5, then angle = arcsin(3/5) which is about 36.87 degrees. The principal range for arcsin and arctan is -90 to 90 degrees and for arccos is 0 to 180 degrees, but inside a right triangle the answer is always a single value between 0 and 90 degrees exclusive, so there is no ambiguity. Calculators and programming languages return arcsin and arccos in radians by default; multiply by 180 / pi (about 57.2958) to get degrees.

Use degrees when the problem is stated in degrees (most geometry homework, surveying, navigation, machining) and radians when you are doing calculus, physics, or any programming work. Spreadsheets like Excel and Google Sheets take radians by default: =SIN(30) returns sin of 30 radians, not 30 degrees. Use =SIN(RADIANS(30)) to get sin of 30 degrees, which is 0.5. JavaScript, Python's math module, and most other standard libraries are the same: Math.sin(Math.PI / 2) equals 1, not Math.sin(90). The conversion is radians = degrees x pi / 180, and degrees = radians x 180 / pi. This calculator accepts angle input in degrees and reports both degrees and radians in the result so you can plug either one directly into the tool you are using next.