In a right triangle, the ratios of the sides depend only on the angles. SOH CAH TOA helps you remember three of them: sine, cosine and tangent.
Try it with your own problem ↓
The steps
- Mark the angle you are working with. From that angle, label the sides: opposite (across from it), adjacent (next to it, not the hypotenuse) and hypotenuse (opposite the right angle).
- Remember: SOH: sin = opposite ÷ hypotenuse. CAH: cos = adjacent ÷ hypotenuse. TOA: tan = opposite ÷ adjacent.
- Pick the ratio that uses the side you know and the side you want.
- Write the equation and solve. For a side, multiply (or divide). For an angle, use the inverse key (sin⁻¹, cos⁻¹, tan⁻¹).
- Make sure your calculator is in degree mode.
A worked example
A 20 ft ladder leans against a wall and makes a 70° angle with the ground. How high does it reach?
- The ladder is the hypotenuse (20 ft). The height is across from the 70° angle, so it is the opposite side.
- Opposite and hypotenuse mean SOH: sin(70°) = height ÷ 20.
- Multiply both sides by 20: height = 20 × sin(70°).
- sin(70°) ≈ 0.93969.
- height ≈ 20 × 0.93969 ≈ 18.79.
The ladder reaches about 18.79 ft up the wall.
Common mistakes
- Calculator in radian mode. If you get a strange answer, check the mode.
- Mislabeling opposite and adjacent. They depend on which angle you picked. Relabel for each angle.
- Using sin when you should use cos. Choose by the sides you have, not by habit.
- Forgetting the inverse for an angle. If you know the sides and want the angle, you need sin⁻¹, cos⁻¹ or tan⁻¹.
Questions people ask
How do I find an angle?
Use the inverse function. If the opposite side is 5 and the hypotenuse is 13, sin(θ) = 5/13, so θ = sin⁻¹(5/13) ≈ 22.62°.
Do SOH CAH TOA work on any triangle?
Only on right triangles. For other triangles use the law of sines and the law of cosines.
What are the special values?
sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1, sin 45° = √2/2, and tan 60° = √3.
Keep learning
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