Tangent (tan)

What is tangent

The tangent of an angle is the ratio of the opposite to the adjacent leg, i.e. sin α / cos α. Geometrically it expresses steepness (slope) – the gradient of a line making that angle with the horizontal.

Domain and properties

The domain is all real numbers except 90° + k·180° (where cos = 0 and tangent is undefined). The range is all real numbers. The period is 180° and the function is odd.

Notable values

tan 0° = 0, tan 30° = √3/3 ≈ 0.577, tan 45° = 1, tan 60° = √3 ≈ 1.732, tan 90° = undefined

Where tangent is used in practice

• Roof pitch, road gradient or ramp slope – a gradient in percent is the tangent of the angle.
• Finding the height of an inaccessible object (a tree, a building) from an angle and a distance.
• Optics and physics – refraction of light, reflection angles.

The unit circle and graphs

The best way to understand trigonometric functions is the unit circle – a circle of radius 1 centred at the origin. Moving along it by an angle α, the coordinates of the point are exactly (cos α, sin α). This gives the fundamental identity sin²α + cos²α = 1 (the Pythagorean theorem for that point). The graph of the function is obtained by plotting the function value against the angle – producing the characteristic wave.

Relationships between the functions

tan α = sin α / cos α  •  cot α = 1 / tan α  •  sin²α + cos²α = 1