What right-angle trigonometry covers
Right-angle trigonometry relates the angles of a right triangle to the ratios of its sides. Engineers use it to find a height from a length and an angle, an angle from two measurements, or the parts of a force along and across a slope. It is a small toolkit that goes a long way.
The sides and the three ratios
Pick one of the two non-right angles, called θ. The side opposite θ is the opposite, the side next to θ (not the longest one) is the adjacent, and the longest side across from the right angle is the hypotenuse. Then:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent
Many people remember this as SOH-CAH-TOA. Note that “opposite” and “adjacent” change if you switch to the other angle, so label the triangle before you start.
Check with a familiar triangle
A triangle with sides 3, 4 and 5 is a right triangle, since 3² + 4² = 9 + 16 = 25 = 5². Take θ as the angle opposite the side of length 3. Then:
- sin θ = 3/5 = 0.6
- cos θ = 4/5 = 0.8
- tan θ = 3/4 = 0.75
The angle itself is about 36.9°. As a consistency check, sin² θ + cos² θ = 0.36 + 0.64 = 1, which is true for any angle. The Pythagorean relation is the quickest test of whether a pair of values is plausible.
Finding an angle from sides
To go from a ratio back to an angle, use the inverse functions (often labelled sin⁻¹, cos⁻¹ and tan⁻¹ or arcsin, arccos and arctan on a calculator).
Example: a ramp rises 1 m over a horizontal run of 12 m. The slope angle satisfies tan θ = 1/12 ≈ 0.0833, so θ = arctan(0.0833) ≈ 4.8°.
Slope is often quoted as a gradient (1 in 12) or a percentage (about 8.3%), not as an angle. For small angles the angle in radians is close to the tangent, which is why a gentle slope and its gradient look so similar. Which form is required depends on your field and local rules, so check the standard that applies.
Finding a length from an angle
A 4 m ladder leans against a wall and makes a 60° angle with the ground.
- Height reached = 4 × sin 60° = 4 × 0.866 ≈ 3.46 m.
- Distance of the base from the wall = 4 × cos 60° = 4 × 0.5 = 2.0 m.
Check with Pythagoras: 3.46² + 2.0² ≈ 11.97 + 4 = 15.97, and 4² = 16. The tiny difference is rounding. This is only a geometry exercise. Real ladder use is governed by safety guidance that you should follow rather than rely on numbers from an article.
Forces on a slope
A weight acting straight down can be split into a part along a slope and a part pressing into it. For a slope at angle θ to the horizontal and a weight W:
- Component along the slope = W × sin θ
- Component perpendicular to the slope = W × cos θ
With W = 100 N and θ = 30°:
- Along the slope: 100 × 0.5 = 50 N
- Perpendicular: 100 × 0.866 ≈ 86.6 N
Check: 50² + 86.6² = 2500 + about 7500 = about 10,000, and 100² = 10,000. The components rebuild the original force.
This is the same splitting used when adding forces in components, which the vector addition refresher builds on. For a steeper slope, the along-slope share rises and the pressing share falls, which matches everyday experience: a steeper ramp is harder to hold something on.
Degrees and radians
Calculators and spreadsheets use either degrees or radians, and mixing them up is among the most common mistakes. A full turn is 360° or 2π radians, so 180° equals π radians (about 3.1416). Test your tool with sin 30°: it should return 0.5 when set to degrees. If you get about −0.988, it is working in radians. Spreadsheet trig functions typically expect radians, so conversion may be needed.
Special values worth knowing
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | about 0.866 | about 0.577 |
| 45° | about 0.707 | about 0.707 | 1 |
| 60° | about 0.866 | 0.5 | about 1.732 |
| 90° | 1 | 0 | undefined |
At 90° the adjacent side shrinks to zero, so the tangent, which divides by it, has no finite value.
Habits for practical work
- Draw and label the triangle, with θ marked, before choosing a formula.
- Choose the ratio that uses the two sides you care about: one known and one wanted.
- Estimate first. A 4 m ladder cannot reach higher than 4 m, so an answer above that is an error.
- Keep sensible rounding, as in significant figures, rather than every digit shown.
- Do a reasonableness check on every result, in the spirit of sanity checks.
Right-angle trigonometry extends to non-right triangles through other rules, but the right-triangle ratios cover a large share of day-to-day needs. These are illustrative calculations; for structural, safety or regulated work, follow the relevant standards and have a qualified person verify the design.