What does SOHCAHTOA stand for?

SOHCAHTOA is a mnemonic for the three primary trigonometric ratios in a right triangle. SOH means Sine equals Opposite over Hypotenuse (sin(θ) = O/H). CAH means Cosine equals Adjacent over Hypotenuse (cos(θ) = A/H). TOA means Tangent equals Opposite over Adjacent (tan(θ) = O/A). Together these let you find any unknown side or angle in a right triangle when you know at least two other values.

How do I use SOHCAHTOA to find a missing side?

First identify which sides you know relative to the angle: the side opposite the angle, the side adjacent to the angle, and the hypotenuse (the longest side, opposite the right angle). Then choose the ratio that connects your known side and your unknown side. For example, if you know the angle is 30 degrees and the hypotenuse is 10, you can use SOH: sin(30°) = Opposite/10, so Opposite = 10 x sin(30°) = 5.000. You can verify using CAH: cos(30°) = Adjacent/10, so Adjacent = 10 x cos(30°) = 8.660.

How do I find a missing angle using SOHCAHTOA?

If you know two sides, use the inverse trigonometric functions. If you know opposite and hypotenuse, use θ = arcsin(O/H). If you know adjacent and hypotenuse, use θ = arccos(A/H). If you know opposite and adjacent, use θ = arctan(O/A). For example, if the opposite side is 5 and the hypotenuse is 10, then θ = arcsin(5/10) = arcsin(0.5) = 30 degrees. The other acute angle in the triangle will be 90 minus 30 = 60 degrees.

SOHCAHTOA Solver

Enter any two known values of a right triangle and this solver will calculate the remaining sides and angles. Choose which two values you know, enter them, and the results update instantly.

SOHCAHTOA stands for: Sine = Opposite / Hypotenuse • Cosine = Adjacent / Hypotenuse • Tangent = Opposite / Adjacent

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Standard Formula Standard trigonometric identities (sin, cos, tan and their inverses). Valid for all right triangles.

1. I Know These Two Values

Select the two known values

Angle (θ) degrees

Hypotenuse (H)

Opposite (O)

Adjacent (A)

2. Triangle Diagram

O = Opposite, A = Adjacent, H = Hypotenuse, θ = angle at bottom-right

Solved Triangle

Angle (θ)
-
degrees

Opposite (O)

side length

Adjacent (A)

side length

Hypotenuse (H)

side length

All Angles

Angle θ-

Complementary angle (90 − θ)-

Right angle90.000°

Sum of angles180.000°

Ratios Used

sin(θ) = O/H-

cos(θ) = A/H-

tan(θ) = O/A-

Formula used-

Worked example:

What Is SOHCAHTOA?

SOHCAHTOA is a mnemonic used to remember the three fundamental trigonometric ratios for a right triangle. Every right triangle has one 90-degree angle and two acute angles. For any chosen acute angle (called theta, or θ), the three sides are named relative to that angle:

Opposite (O): the side directly opposite the angle θ
Adjacent (A): the side that forms the angle θ alongside the hypotenuse
Hypotenuse (H): the longest side, always opposite the 90-degree angle

The three ratios are:

Name	Abbreviation	Formula	Inverse (to find angle)
Sine	sin	sin(θ) = O / H	θ = arcsin(O / H)
Cosine	cos	cos(θ) = A / H	θ = arccos(A / H)
Tangent	tan	tan(θ) = O / A	θ = arctan(O / A)

How to Use This Solver

Select which two values you already know from the dropdown. The possible combinations are:

Angle + Hypotenuse: find O using SOH (O = H × sin(θ)) and A using CAH (A = H × cos(θ))
Angle + Opposite: find H using SOH (H = O / sin(θ)) and A using TOA (A = O / tan(θ))
Angle + Adjacent: find H using CAH (H = A / cos(θ)) and O using TOA (O = A × tan(θ))
Opposite + Hypotenuse: find θ using arcsin(θ = arcsin(O / H)) and A using Pythagoras (A = √(H² − O²))
Adjacent + Hypotenuse: find θ using arccos (θ = arccos(A / H)) and O using Pythagoras (O = √(H² − A²))
Opposite + Adjacent: find θ using arctan (θ = arctan(O / A)) and H using Pythagoras (H = √(O² + A²))

Worked Example

Given: angle θ = 30° and hypotenuse H = 10.

Opposite: O = H × sin(30°) = 10 × 0.5000 = 5.000
Adjacent: A = H × cos(30°) = 10 × 0.8660 = 8.660
Complementary angle: 90 − 30 = 60°
Verify with Pythagoras: O² + A² = 25 + 75 = 100 = H² ✓

This matches the default output of the calculator above.

Common Angle Values

Angle (θ)	sin(θ)	cos(θ)	tan(θ)
0°	0.000	1.000	0.000
30°	0.500	0.866	0.577
45°	0.707	0.707	1.000
60°	0.866	0.500	1.732
90°	1.000	0.000	undefined

Tips and Common Mistakes

SOHCAHTOA only works for right triangles. For triangles without a 90-degree angle, use the Sine Rule or Cosine Rule.
The angle θ must be one of the acute angles, not the right angle.
Make sure your calculator is set to degrees (not radians) when computing sin, cos, and tan manually.
The hypotenuse is always the longest side and is always opposite the right angle.
If you know all three sides and no angle, use arcsin, arccos, or arctan to recover the angle.

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Method: Standard trigonometric ratios (SOH = sin(θ) = O/H, CAH = cos(θ) = A/H, TOA = tan(θ) = O/A) and their inverses (arcsin, arccos, arctan). The Pythagorean theorem (H² = O² + A²) is used as a cross-check. Implemented using JavaScript's built-in Math.sin, Math.cos, Math.tan, Math.asin, Math.acos, and Math.atan2 functions, which operate in radians internally with conversion to and from degrees.

This solver assumes a standard right triangle. All input angles must be between 0 and 90 degrees (exclusive). Side lengths must be positive. Rounding is to three decimal places. For ambiguous cases (for example, where a given opposite side is longer than the hypotenuse), the solver will display an error.