Tangent Function
The tangent function is defined as the ratio of the sine to the cosine of an angle (tan θ = sin θ / cos θ), or equivalently as the ratio of the opposite side to the adjacent side in a right triangle. Unlike sine and cosine, the tangent function has a period of π and is undefined at θ = π/2 + nπ (where n is any integer) because cosine equals zero at those points, producing vertical asymptotes. Tangent is fundamental in calculating slopes of lines, angles of elevation and depression, and in integral calculus substitutions.
Key Formula
tan(θ) = sin(θ) / cos(θ) = opposite / adjacent
LaTeX: \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\text{opposite}}{\text{adjacent}}
| Symbol | Meaning | Unit |
|---|---|---|
| θ | angle measured from the positive x-axis | radians or degrees |
| sin(θ) | sine of the angle | dimensionless |
| cos(θ) | cosine of the angle (must be non-zero) | dimensionless |
| opposite | side opposite angle θ in a right triangle | length units |
| adjacent | side adjacent to angle θ (not the hypotenuse) | length units |
Worked Example
Problem
From a point 50 m from the base of a vertical tower, the angle of elevation to the top is 38°. Find the height of the tower.
Solution
Step 1: The opposite side is the tower height h; the adjacent side is the horizontal distance = 50 m. Step 2: tan(38°) = h / 50. Step 3: h = 50 × tan(38°) = 50 × 0.7813 = 39.07 m.
Answer
The tower is approximately 39.1 m tall.
Tangent Values and Asymptote Behaviour
| Angle (degrees) | Angle (radians) | tan(θ) | Behaviour |
|---|---|---|---|
| 0° | 0 | 0 | Crosses x-axis |
| 30° | π/6 | 0.5774 | 1/√3 |
| 45° | π/4 | 1.0000 | Slope of 45° line |
| 60° | π/3 | 1.7321 | √3 |
| 89° | ≈ π/2 − 0.017 | 57.29 | Approaches +∞ |
| 90° | π/2 | Undefined | Vertical asymptote |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Sine Function
The sine function is a fundamental trigonometric function defined for an angle θ in a right triangle as the ratio of the length of the side opposite the angle to the length of the hypotenuse, extended to all real numbers via the unit circle. It is a periodic function with period 2π, amplitude 1, and range [−1, 1], producing a smooth oscillating wave. Sine is essential in modelling wave phenomena including sound, light, alternating current, and simple harmonic motion.
Cosine Function
The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle, and extended to all real numbers as the x-coordinate of a point on the unit circle at angle θ from the positive x-axis. Like sine, it is periodic with period 2π and range [−1, 1], but is phase-shifted by π/2 relative to sine (cos θ = sin(θ + π/2)). Cosine is widely used in Fourier analysis, wave optics, mechanical vibrations, and calculating dot products of vectors.
Radian
A radian is the SI unit of angular measure defined as the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle. Since the full circumference is 2πr, one complete revolution equals 2π radians, giving the exact conversion 180° = π radians. Radians are the natural unit for trigonometry and calculus because they make derivative formulas for trigonometric functions simple (d/dθ sin θ = cos θ holds only when θ is in radians).
From the Latin tangens ("touching"), present participle of tangere ("to touch"). The term was introduced by Danish mathematician Thomas Fincke in his 1583 work Geometria rotundi, because the tangent can be represented geometrically as the length of a line that "touches" the unit circle at one point.