SOHCAHTOA
SOHCAHTOA is the mnemonic for the three basic trig ratios: sin = O/H, cos = A/H, tan = O/A.
Essential for solving right triangles.
The Formulas
SOH: sin(θ) = Opposite / Hypotenuse
CAH: cos(θ) = Adjacent / Hypotenuse
TOA: tan(θ) = Opposite / Adjacent
SOHCAHTOA is a mnemonic for remembering the three basic trigonometric ratios.
These ratios only apply to right triangles (triangles with a 90° angle).
Variables
| Term | Meaning |
|---|---|
| θ | The angle you are working with (not the 90° angle) |
| Opposite | The side across from angle θ |
| Adjacent | The side next to angle θ (not the hypotenuse) |
| Hypotenuse | The longest side (opposite the 90° angle) |
How to Choose the Right Ratio
- Identify the angle you are working with
- Label the three sides as Opposite, Adjacent, and Hypotenuse
- Pick the ratio that uses the two sides you know (or the side you know and the one you need)
Example 1
In a right triangle, the angle is 35° and the hypotenuse is 20. Find the opposite side.
We know: angle = 35°, Hypotenuse = 20, we want Opposite
Use SOH: sin(35°) = Opposite / 20
Opposite = 20 × sin(35°) = 20 × 0.5736
Opposite ≈ 11.47
Example 2
A tree casts a shadow 15 m long. The angle of elevation to the top of the tree is 53°. How tall is the tree?
Adjacent = 15 m (shadow), angle = 53°, we want Opposite (tree height)
Use TOA: tan(53°) = Opposite / 15
Opposite = 15 × tan(53°) = 15 × 1.3270
The tree is approximately 19.91 m tall
When to Use It
Use SOHCAHTOA when:
- Working with a right triangle and you know one angle and one side
- Finding the height of objects using angles of elevation
- Calculating distances using angles of depression
- Solving any right triangle problem (for non-right triangles, use the Law of Sines or Law of Cosines)
Key Notes
- Mnemonic: SOH-CAH-TOA: sin θ = Opposite/Hypotenuse; cos θ = Adjacent/Hypotenuse; tan θ = Opposite/Adjacent. These ratios apply only in right triangles and only for the acute angles (not the 90° angle itself). The hypotenuse is always opposite the right angle.
- Which angle determines "opposite" and "adjacent": Opposite and adjacent are relative to the angle being evaluated, not fixed sides of the triangle. For angle A, the opposite side is the one not touching angle A; the adjacent side is the one touching A that is not the hypotenuse.
- Finding angles with inverse trig: If sinθ = 0.5, then θ = arcsin(0.5) = 30°. Always check the context — arcsin returns values in [−90°, 90°], so the supplementary angle (180° − 30° = 150°) may also be a valid solution depending on the triangle's configuration.
- Extension to the unit circle: For any angle (not just acute), sin θ and cos θ are the y and x coordinates of the corresponding point on the unit circle. SOH-CAH-TOA is the special case for acute angles in a right triangle — consistent with but less general than the unit circle definition.
- Applications: SOHCAHTOA is used to find unknown sides or angles in right triangles: building heights (shadow and angle of elevation), GPS triangulation, ramp slope angles, staircase design (rise/run angle), navigation bearings, and as the entry point to all trigonometric reasoning in physics and engineering.