tangente

Tangente, or the tangent function, is a trigonometric function defined for real angles θ by tan θ = sin θ / cos θ, provided cos θ ≠ 0. It is periodic with period π and has vertical asymptotes at θ = π/2 + kπ (k an integer). The function is odd, meaning tan(−θ) = −tan θ, and its range is all real numbers.

The domain of tangente consists of all real numbers except the points where cos θ = 0; at these points the function is undefined. Its graph consists of repeating S-shaped branches that rise from −∞ to +∞ between consecutive asymptotes. The derivative of tangente is sec² θ, equivalently 1 / cos² θ, and its integral is ∫ tan θ dθ = −ln|cos θ| + C = ln|sec θ| + C.

Key trigonometric identities involve tangente: tan(α + β) = (tan α + tan β) / (1 − tan α tan β); tan(2θ) = 2 tan θ / (1 − tan² θ). The Pythagorean identity 1 + tan² θ = sec² θ is a direct consequence of sin² θ + cos² θ = 1.

Geometrically, tangente can be interpreted as the slope of the line through the origin making angle θ with the positive x-axis. On the unit circle, it also relates to the intersection of that line with the tangent line at (1,0), where its height equals tan θ.

Tangente appears in various applications across mathematics, physics, and engineering, including slope calculations, solving trigonometric equations, and in calculus as a basic function with well-known derivatives and integrals.