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.
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.
Practice Questions
- Find the opposite side: adjacent = 8 cm, θ = 55°.
- Find the adjacent side: opposite = 12 cm, θ = 30°.
- Find angle θ: opposite = 9 cm, adjacent = 5 cm.
- A flagpole casts a shadow 20 m long. The angle of elevation to the top is 28°. How tall is the flagpole?
- From a window 30 m above the ground the angle of depression to a parked car is 40°. How far is the car from the base of the building?