Right-triangle trigonometry
The idea
Right-triangle trigonometry converts angle information into length information. In a right triangle, the ratios between side lengths depend only on the acute angle — a direct consequence of similarity, since all right triangles sharing that angle are scaled copies of one another. Naming the sides relative to the chosen angle θ — opposite, adjacent, hypotenuse — defines the three ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. From one known side and one acute angle, every other measurement of the triangle is within reach.
The practical workflow: sketch the triangle, mark the angle, label which sides are known and wanted, then pick the one ratio that connects them. To recover an angle from two sides, use the inverse functions sin⁻¹, cos⁻¹, or tan⁻¹. The classic mistake is labeling sides in absolute terms — opposite and adjacent are relative to the angle in use, and they swap when you switch to the other acute angle. A close second is a calculator left in the wrong angle mode; if the output makes no sense against your sketch, check the mode before anything else.
Worked example
From a point on level ground 40 m from the base of a vertical tower, the angle of elevation to the top is 52°. Find the height of the tower and the straight-line distance to its top, each to one decimal place.
- Sketch the right triangle: the 40 m ground distance is adjacent to the 52° angle, the height h is opposite it, and the line of sight is the hypotenuse.
- Height connects opposite and adjacent, which is tangent's job: tan 52° = h/40, so h = 40 × tan 52° ≈ 40 × 1.2799 ≈ 51.2 m.
- The line-of-sight distance d pairs adjacent with hypotenuse, so use cosine: cos 52° = 40/d, giving d = 40/cos 52° ≈ 40/0.6157 ≈ 65.0 m.
- Cross-check with the Pythagorean theorem: 40² + 51.2² = 1600 + 2621.4 = 4221.4, and √4221.4 ≈ 65.0 — two independent routes land on the same hypotenuse.
Answer. The tower is about 51.2 m tall, and its top is about 65.0 m from the observation point.
Check your understanding
- Why do the trigonometric ratios of an angle stay the same regardless of the triangle's size, and which earlier geometric idea guarantees it?
- How do you choose among sine, cosine, and tangent for a given problem before reaching for the calculator?
- What happens to the tangent ratio as the angle of elevation climbs toward 90°, and what does that mean about the tower in the picture?
- How are sin θ and cos(90° − θ) related, and how does the side-labeling of the two acute angles explain it?
Build the foundations first
Right-triangle trigonometry builds on these concepts. If any feel shaky, start there.