In a right triangle, the tangent of an angle is the opposite side divided by the adjacent side. For a 45-degree angle, tan 45° = 1 because the opposite and adjacent sides are equal. The function is undefined at 90 degrees and its odd multiples.

Tangent is periodic with period 180 degrees, or π radians. Its graph has vertical asymptotes where the function is undefined. Along with sine and cosine, it is one of the three main trigonometric functions.

Tangent appears in geometry, physics, and engineering. It relates angles to slopes: the slope of a line is the tangent of its angle of inclination. In calculus, the derivative of tan x is sec²x. The function is essential for problems involving angles, distances, and rates.

• Ratio of opposite to adjacent side
• Period π; undefined at 90°
• Relates angle to slope
• Derivative is sec²x