SOHCAHTOA Solver
This SOHCAHTOA solver works out the missing sides and angles of any right triangle using the sine, cosine and tangent ratios. Right triangles have three sides, the hypotenuse (the longest, opposite the right angle), the opposite and the adjacent, named relative to one of the two acute angles, and if you know any two of these four values you can find the rest. Choose which two values you already know from the dropdown, angle and hypotenuse, angle and opposite, angle and adjacent, opposite and hypotenuse, adjacent and hypotenuse, or opposite and adjacent, then enter those figures. The solver instantly returns the angle theta, the opposite side, the adjacent side and the hypotenuse, plus a breakdown showing the complementary angle, the sine, cosine and tangent values used, which formula was applied, and a worked example that checks your answer against Pythagoras' theorem. A triangle diagram beside the inputs shows how each side and angle is labelled, so there is no confusion about which is which. It suits maths homework, trades, technical drawing, surveying, and any job needing a triangle's dimensions from limited information. Results are rounded to three decimal places, and the known angle must be between 0 and 90 degrees. If your inputs are impossible, such as an opposite side longer than the hypotenuse, the solver flags an error instead of a result.
1. I Know These Two Values
2. Triangle Diagram
O = Opposite, A = Adjacent, H = Hypotenuse, θ = angle at bottom-right
All Angles
Ratios Used
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.