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Geometry guide

Right triangle trigonometry: SOH CAH TOA

Sine, cosine and tangent in right triangles, how to find missing sides and angles, special triangles, and angle of elevation problems step by step.

Trigonometry connects the angles of a right triangle to the lengths of its sides. With three ratios and a calculator, you can find heights you cannot measure directly, such as a building, a tree or a ramp's slope. This guide covers the ratios, how to solve for sides and angles, and the word problems that use them.

Naming the sides

In a right triangle, pick one of the acute angles and call it θ. Then:

  • The hypotenuse is the longest side, opposite the right angle.
  • The opposite side is across from θ.
  • The adjacent side is next to θ and is not the hypotenuse.

Opposite and adjacent depend on which angle you choose; the hypotenuse never changes.

The three ratios

Remember SOH CAH TOA:

  • sin θ = opposite / hypotenuse
  • cos θ = adjacent / hypotenuse
  • tan θ = opposite / adjacent

These ratios depend only on the angle, not the size of the triangle, because all right triangles with the same angle are similar.

Worked example 1: find a missing side

A right triangle has a 35° angle and a hypotenuse of 12. Find the side opposite the 35° angle.

  1. You know the hypotenuse and want the opposite side, so use sine.
  2. sin 35° = x / 12, so x = 12 sin 35° ≈ 12 × 0.574 ≈ 6.88.

Make sure your calculator is in degree mode, not radians.

Worked example 2: find a missing angle

A ramp rises 1.5 m over a horizontal distance of 6 m. What angle does it make with the ground?

  1. You know the opposite (1.5) and adjacent (6), so use tangent: tan θ = 1.5 / 6 = 0.25.
  2. Use the inverse: θ = tan⁻¹(0.25) ≈ 14.0°.

Angles of elevation and depression

The angle of elevation is measured upward from the horizontal to an object; the angle of depression is measured downward. Both are measured from a horizontal line, never from the vertical.

Worked example 3: height of a tree

From 20 m away, the angle of elevation to the top of a tree is 40°. Your eyes are 1.6 m above the ground. How tall is the tree?

  1. tan 40° = h / 20, so h = 20 tan 40° ≈ 20 × 0.839 ≈ 16.8 m above eye level.
  2. Add your eye height: 16.8 + 1.6 = 18.4 m.

Special right triangles

  • 45-45-90: sides in the ratio 1 : 1 : √2. If each leg is 5, the hypotenuse is 5√2.
  • 30-60-90: sides in the ratio 1 : √3 : 2, with the shortest side opposite 30°. If the short leg is 4, the long leg is 4√3 and the hypotenuse is 8.

These let you give exact answers, such as sin 30° = ½ and cos 45° = √2/2, without a calculator.

Checking your answer

  • The hypotenuse must be the longest side.
  • The two acute angles must add to 90°.
  • Use the Pythagorean theorem, a² + b² = c², to confirm all three sides fit together.

Common mistakes

  • Calculator in radian mode.
  • Labeling opposite and adjacent from the wrong angle.
  • Using SOH CAH TOA on a triangle with no right angle. For those, use the law of sines or cosines.
  • Forgetting to add the observer's height in elevation problems.

Practice questions

  1. In a right triangle, the adjacent side to a 50° angle is 8. Find the hypotenuse.
  2. A 10 m ladder leans against a wall, making a 70° angle with the ground. How high up the wall does it reach?
  3. Find θ if sin θ = 0.6.
  4. A 30-60-90 triangle has a hypotenuse of 14. Find the short leg.

Answers: (1) 8 / cos 50° ≈ 12.4. (2) 10 sin 70° ≈ 9.40 m. (3) θ = sin⁻¹(0.6) ≈ 36.9°. (4) 7.

Last reviewed 2026-10-23 by the BrainBoost team.

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