Tangent - Opposite Over Adjacent
Tangent is the third of the three primary trigonometric ratios. It compares the side opposite an angle to the side adjacent to it in a right triangle. The tangent ratio is the workhorse of problems that involve slopes, gradients, and angles of elevation or depression.
The word “tangent” comes from the Latin tangere, meaning “to touch”, because the ratio was originally defined using a line segment that touches a circle at a single point – the same geometric idea used today for a tangent line to a curve. The Danish mathematician Thomas Fincke is credited with introducing the term tangens in his 1583 book Geometria Rotundi, one of the first works to organise the trigonometric ratios into the systematic form still taught today.
Tangent is the ratio engineers reach for whenever they need a slope: road gradients are often quoted as a percentage that is really 100 × tan(θ), and the “rise over run” used to describe wheelchair ramps and staircases is exactly the tangent of the incline angle.
Definition
For an acute angle θ in a right triangle:
tan(θ) = Opposite ÷ Adjacent
Memory aid: TOA – Tangent = Opposite over Adjacent (the last part of SOH CAH TOA).
Also: tan(θ) = sin(θ) ÷ cos(θ).
Key Values of Tangent
| Angle (θ) | tan(θ) | Exact value |
|---|---|---|
| 0° | 0 | 0 |
| 30° | ≈ 0.577 | 1/√3 |
| 45° | 1 | 1 |
| 60° | ≈ 1.732 | √3 |
| 90° | undefined | ∞ |
Note: tan(90°) is undefined because the adjacent side shrinks to zero – you would be dividing by zero.
Angle of Elevation and Depression
Angle of elevation – the angle you look up from the horizontal to see an object.
Angle of depression – the angle you look down from the horizontal.
In both cases, the tangent ratio links the height (opposite) to the horizontal distance (adjacent).
Worked Examples
Opposite = 10 × tan(40°) = 10 × 0.8391 = 8.39 cm.
Height = 15 × tan(35°) = 15 × 0.7002 ≈ 10.5 m.
tan(25°) = 80 / distance. Distance = 80 / tan(25°) = 80 / 0.4663 ≈ 171.6 m.
tan(θ) = 6/6 = 1. θ = tan−¹(1) = 45°.
SOH CAH TOA Summary
| Ratio | Formula | Memory |
|---|---|---|
| Sine | Opposite / Hypotenuse | SOH |
| Cosine | Adjacent / Hypotenuse | CAH |
| Tangent | Opposite / Adjacent | TOA |
Key Takeaways
- tan(θ) = Opposite / Adjacent (TOA).
- tan(45°) = 1 because opposite = adjacent in a 45–45–90 triangle.
- tan(90°) is undefined – it tends to infinity.
- Use tan for angles of elevation and depression problems where height and horizontal distance are involved.