Trigonometry7 min read
How to Find Missing Angles in a Right Triangle: 3 Easy Methods
Learn how to find missing angles in right triangles with 3 easy methods. Master trigonometry basics with step-by-step examples and practical applications.
Enter Any 2 Values
Remember:
SOH sin = Opp / Hyp
CAH cos = Adj / Hyp
TOA tan = Opp / Adj
Results
Enter at least 2 values to solve the triangle
Opposite
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Adjacent
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Hypotenuse
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Angle θ (deg)
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Other angle (deg)
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sin θ
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cos θ
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tan θ
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Triangle
SOHCAHTOA Explained
SOHCAHTOA is a mnemonic that captures the three primary trigonometric ratios for a right triangle. It's one of the first things every trigonometry student memorizes — and once you've internalized it, every right-triangle problem reduces to the same workflow: identify the two sides involved, pick the matching ratio, and solve.
The three ratios
SOH — sin(θ) = opposite ÷ hypotenuse
CAH — cos(θ) = adjacent ÷ hypotenuse
TOA — tan(θ) = opposite ÷ adjacent
Identifying the sides
For any chosen acute angle θ in a right triangle:
- Hypotenuse — always the longest side, opposite the 90° angle. This never changes.
- Opposite — the leg across from θ. It does not touch θ at all.
- Adjacent — the leg next to θ that isn't the hypotenuse.
Worked example
Suppose you have a right triangle where θ = 30° and the hypotenuse is 10. To find the opposite side, pick the ratio that uses opposite and hypotenuse — that's SOH. So:
sin(30°) = opp / 10
opp = 10 × sin(30°) = 10 × 0.5 = 5
opp = 10 × sin(30°) = 10 × 0.5 = 5
To find the adjacent side, use CAH: adj = 10 × cos(30°) ≈ 8.66. To verify, the Pythagorean theorem gives 5² + 8.66² ≈ 100 = 10². ✓
Finding angles with inverse trig
When you know two sides and want the angle, use the inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹). For instance, if opposite = 3 and adjacent = 4:
tan(θ) = 3 / 4 = 0.75
θ = arctan(0.75) ≈ 36.87°
θ = arctan(0.75) ≈ 36.87°
For a deeper dive into the inverse functions, see our inverse trig calculator.
When SOHCAHTOA doesn't apply
SOHCAHTOA only works in right triangles. For oblique (non-right) triangles, switch to the Law of Sines or the Law of Cosines.
Related calculators
- Sin Cos Tan Calculator — get all six trig values for any angle
- Inverse Trig Calculator — find an angle from a ratio
- Right Triangle Calculator — solve any right triangle from two values
- Pythagorean Theorem Calculator
Frequently Asked Questions
What does SOHCAHTOA stand for?
SOHCAHTOA is a mnemonic for the three primary trig ratios in a right triangle: SOH = Sine equals Opposite over Hypotenuse, CAH = Cosine equals Adjacent over Hypotenuse, TOA = Tangent equals Opposite over Adjacent. It helps you remember which sides each function uses.
When do I use sin vs cos vs tan?
Use sin when you have (or want) the opposite side and the hypotenuse. Use cos when you have the adjacent side and the hypotenuse. Use tan when you have only the two legs (opposite and adjacent), with no hypotenuse involved. Pick the function whose two sides match what you know.
What's the difference between opposite and adjacent?
It depends on which acute angle you're working from. The "opposite" side is the leg directly across from your angle of interest (not touching it). The "adjacent" side is the leg next to your angle that ISN'T the hypotenuse. The hypotenuse is always the longest side, opposite the right angle.
How do I find an angle from two sides?
Use an inverse trig function. If you know opposite and hypotenuse: angle = arcsin(opp/hyp). Adjacent and hypotenuse: angle = arccos(adj/hyp). Opposite and adjacent: angle = arctan(opp/adj). On a calculator, these are usually labelled sin⁻¹, cos⁻¹, and tan⁻¹.
Does SOHCAHTOA only work for right triangles?
Yes — these ratios are defined only for right triangles. For non-right (oblique) triangles, use the Law of Sines or Law of Cosines instead. You can also split an oblique triangle into two right triangles by drawing an altitude, then apply SOHCAHTOA to each.
What if I only know one side and want to find the others?
One side alone isn't enough — you need at least two pieces of information. Either a second side, or one of the acute angles. With one side and one angle (besides the 90°), this calculator can find every other measurement.
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