Naming the three sides
A right triangle has one 90° angle. The side across from it is the hypotenuse, always the longest side. The other two sides are named from whichever acute angle you are working with, usually called θ (theta). The opposite side is the one that does not touch θ; the adjacent side touches θ and is not the hypotenuse. Switch to the other acute angle and opposite and adjacent swap places, while the hypotenuse stays put. Getting this naming right is most of the work; the rest is division.
The three ratios
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
Each ratio is a plain number with no unit, because it divides a length by a length. Sine and cosine are never larger than 1, since the hypotenuse is the longest side. Tangent can be any size: it is the slope of the hypotenuse, rise over run, and it also equals sin θ ÷ cos θ. The ratios depend only on the angle, not on the size of the triangle. Double every side and the quotients do not change, which is why one table of values serves every right triangle ever drawn. The scientific calculator has all three functions and their inverses; make sure it is in degree mode for the problems below.
Values for 0°, 30°, 45°, 60° and 90°
Five angles come up so often that their exact forms are worth knowing. They come from two triangles: half of a square gives 45°, half of an equilateral triangle gives 30° and 60°. The decimals are rounded to four places.
| Angle | Radians | sin | sin ≈ | cos | cos ≈ | tan | tan ≈ |
|---|---|---|---|---|---|---|---|
| 0° | 0 | 0 | 0.0000 | 1 | 1.0000 | 0 | 0.0000 |
| 30° | π/6 | 1/2 | 0.5000 | √3/2 | 0.8660 | 1/√3 | 0.5774 |
| 45° | π/4 | √2/2 | 0.7071 | √2/2 | 0.7071 | 1 | 1.0000 |
| 60° | π/3 | √3/2 | 0.8660 | 1/2 | 0.5000 | √3 | 1.7321 |
| 90° | π/2 | 1 | 1.0000 | 0 | 0.0000 | none | undefined |
Read the table in both directions: the sine column runs upward from 0 to 1 while the cosine column runs down from 1 to 0, and sin 30° equals cos 60°. That is the general rule sin θ = cos (90° − θ), because the opposite side of one acute angle is the adjacent side of the other. Tangent crosses 1 at 45°, where opposite and adjacent are equal, and has no value at 90°.
Which ratio to use
Look at the side you know and the side you want. The pair of sides decides the function, and the equation is solved for whichever side is missing.
| You know | You want | Ratio | Equation |
|---|---|---|---|
| Hypotenuse | Opposite | sin | opposite = hypotenuse × sin θ |
| Hypotenuse | Adjacent | cos | adjacent = hypotenuse × cos θ |
| Adjacent | Opposite | tan | opposite = adjacent × tan θ |
| Opposite | Hypotenuse | sin | hypotenuse = opposite ÷ sin θ |
| Adjacent | Hypotenuse | cos | hypotenuse = adjacent ÷ cos θ |
| Opposite | Adjacent | tan | adjacent = opposite ÷ tan θ |
If you know two sides and want the third, you do not need trigonometry at all: a² + b² = c² does it, and the Pythagorean theorem calculator solves for any one of the three. If you know two sides and want an angle, see the inverse functions in problem 2.
Problem 1: find a side
A ramp climbs at 15° over a horizontal run of 20 ft. How high does it rise, and how long is the ramp surface?
- Name the sides from the 15° angle. The run of 20 ft touches the angle, so it is the adjacent side. The rise is across from the angle: opposite. The ramp surface is the hypotenuse.
- Rise: known adjacent, wanted opposite, so tangent. tan 15° = rise ÷ 20, so rise = 20 × tan 15° = 20 × 0.2679 = 5.36 ft.
- Ramp length: known adjacent, wanted hypotenuse, so cosine. cos 15° = 20 ÷ length, so length = 20 ÷ cos 15° = 20 ÷ 0.9659 = 20.71 ft.
Check with the Pythagorean theorem: 20² + 5.36² = 428.72, and the square root of that is 20.71, matching the ramp length. The triangle calculator gives the same answers from the angle and one side, and draws the triangle to scale.
Problem 2: find an angle with the inverse functions
A roof has a 4 in 12 pitch: it rises 4 inches for every 12 inches of horizontal run. What is the angle of the roof?
- Rise is opposite, run is adjacent, so tangent: tan θ = 4 ÷ 12 = 0.3333.
- To get from the ratio back to the angle, apply the inverse tangent, written tan⁻¹ or arctan: θ = tan⁻¹(0.3333) = 18.4°.
The inverse functions undo the ratios: sin⁻¹ turns a sine value into its angle, cos⁻¹ and tan⁻¹ do the same for the other two. On most calculators they sit on the same keys as sin, cos and tan, reached with a shift or 2nd key. Note that tan⁻¹ means the inverse function here, not 1 ÷ tan. A 12 in 12 roof, the steepest common pitch, works out to tan⁻¹(1) = 45.0°, which is the 45° entry in the table.
Problem 3: a ladder against a wall
A 20 ft ladder stands with its foot 5 ft from the wall, which is the quarter-length setup the OSHA construction rule calls for. What angle does it make with the ground, and how high up the wall does it reach?
- The ladder is the hypotenuse, the 5 ft on the ground is adjacent to the ground angle, and the height on the wall is opposite.
- Angle: known adjacent and hypotenuse, so cosine. cos θ = 5 ÷ 20 = 0.2500, so θ = cos⁻¹(0.2500) = 75.5°.
- Height: known hypotenuse, wanted opposite, so sine. height = 20 × sin 75.5° = 19.36 ft.
Check: √(5² + 19.36²) = 20.00 ft, the ladder length. The angle is the same for every ladder set up at one quarter of its length, because cos θ = 0.25 no matter how long the ladder is. The guide to the 4 to 1 ladder rule has the heights and foot distances for common ladder lengths.
Degrees vs radians
A full turn is 360° or 2π radians, so 180° = π radians and 1 radian = 180° ÷ π = 57.30°. One degree is π ÷ 180 = 0.01745 radians. The table above lists both, and the pattern is simple: 30° is π/6, 45° is π/4, 60° is π/3, 90° is π/2. Programming languages work in radians: JavaScript's Math.sin, for example, takes its argument in radians, as its documentation states. Most calculators can be switched between the two, which is the usual cause of a trig answer that is off by a lot rather than a little. sin 30° is 0.5; sin of 30 radians is -0.9880. The angle converter switches degrees, radians, gradians and minutes of arc in either direction.
degrees = radians × 180 ÷ π
Triangles without a right angle
The three ratios only exist in a right triangle, because they are built on the hypotenuse. For any other triangle, the law of sines and the law of cosines take over; the triangle calculator's page explains both, works a side-angle-side example and covers the ambiguous case where two sides and an angle fit two different triangles. The law of cosines is the Pythagorean theorem with a correction term, and it collapses to a² + b² = c² when the angle is 90°, which is one way to see that everything on this page is the right-angle special case of a larger set of rules.
The memory aid, and the mistakes it does not prevent
SOH CAH TOA packs the three definitions into one word: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. It settles which sides go into each ratio. It does not settle three things that cause most wrong answers:
- Which angle you are using. Opposite and adjacent are relative to θ. Mark the angle on the sketch before naming the sides.
- Degree or radian mode. Check the calculator's indicator, or convert the angle, before pressing sin.
- Rounding too early. Keep four decimals in the ratio and round only the final length or angle. In the ladder problem, rounding cos⁻¹(0.25) to 76° first would give a height of 19.41 ft instead of 19.36.
Frequently asked questions
What does SOHCAHTOA stand for?
Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. The three syllables are the three ratios in a right triangle, each as the first letter of the function followed by the two sides it divides. Say it as "so-ka-toe-ah".
Which side is opposite and which is adjacent?
It depends on the angle you are working from. The opposite side is the one across the triangle from that angle, the one that does not touch it. The adjacent side touches the angle and is not the hypotenuse. The hypotenuse is always the longest side, across from the right angle, whichever angle you pick.
How do I find an angle instead of a side?
Use the inverse functions: sin⁻¹, cos⁻¹ and tan⁻¹, also written arcsin, arccos and arctan. Divide the two sides you know to get the ratio, then apply the inverse of the matching function. For a rise of 4 over a run of 12, tan θ = 4 ÷ 12, so θ = tan⁻¹(0.3333), about 18.4 degrees.
Why does my calculator give a different answer?
Almost always because it is set to radians while you typed degrees, or the other way round. sin 30 in degree mode is 0.5; in radian mode it is about −0.988. Check the mode indicator before every trig problem, or convert the angle first.
Does SOHCAHTOA work for any triangle?
No, only for right triangles, because the three ratios are defined with the hypotenuse and the right angle. For a triangle without a right angle you either drop a perpendicular to split it into two right triangles, or use the law of sines and the law of cosines, which work for any triangle.
Why is tan 90° undefined?
Tangent is opposite over adjacent, or equivalently sin over cos. At 90 degrees the cosine is 0, and dividing by 0 has no value. Geometrically, a right triangle cannot have a second 90-degree angle, so there is no triangle to measure. As the angle approaches 90 degrees the tangent grows without limit.
Sources
- 29 CFR 1926.1053(b)(5)(i), OSHA construction ladder rules (a non-self-supporting ladder is used at an angle where the horizontal distance from the top support to the foot is approximately one quarter of the working length, the setup in problem 3)
- MDN Web Docs: Math.sin() (the argument is an angle in radians, the convention of programming languages mentioned under degrees vs radians)