📚 The tangent function | 正切函数
The tangent function, denoted as tan x, is one of the fundamental trigonometric functions. Defined as the ratio of sine to cosine, it plays a crucial role in geometry, calculus, and physics. In IB Mathematics, understanding its properties, graph, and calculus operations is essential for solving trigonometric equations and modelling periodic phenomena. Mastering the tangent function opens the door to deeper topics such as inverse functions, differential equations, and harmonic analysis.
正切函数，记作 tan x，是最基本的三角函数之一。它定义为正弦与余弦的比值，在几何、微积分和物理学中起着至关重要的作用。在IB数学中，理解正切函数的性质、图像和微积分运算是解三角方程和建立周期现象模型的基础。掌握正切函数为深入学习反函数、微分方程和谐波分析等更深入的主题打开了大门。
1. Definition and Basic Ratio | 定义与基本比值
For any angle θ (measured in radians or degrees), the tangent function is defined as tan θ = sin θ / cos θ, provided cos θ ≠ 0. In a right‑ e to vertical asymptotes in its graph. Unlike sine and cosine, the range of tan x is all real numbers (ℝ), while sin x and cos x are bounded between –1 and 1. This unbounded behaviour is a key feature that distinguishes tan x from the other circular functions.
对于任意角 θ（以弧度或度为单位），正切函数定义为 tan θ = sin θ / cos θ，前提是 cos θ ≠ 0。在直角三角形中，这一比值等于对边长度除以邻边长度。由于分母可能趋近于零，每当 cos θ = 0 时函数无定义，这就导致图像中出现垂直渐近线。与正弦和余弦不同，tan x 的值域是全体实数（ℝ），而 sin x 和 cos x 被限制在 –1 到 1 之间。这种无界行为是 tan x 区别于其他圆函数的一个重要特征。
2. The Unit Circle Interpretation | 单位圆中的解释
On the unit circle, any point is described by coordinates (cos θ, sin θ). The tangent of θ then represents the slope of the line segment joining the origin to that point — equivalently, the y‑coordinate divided by the x‑coordinate. Visually, tan θ can be seen as the length of the vertical segment that is tangent to the unit circle at (1,0) and intersected by the terminal ray. This geometric representation explains the sign pattern: tan θ is positive in quadrants I and III (where sin and cos share the same sign) and negative in quadrants II and IV (opposite signs).
在单位圆上，任意点用坐标 (cos θ, sin θ) 描述。θ 的正切值就表示从原点到该点的线段斜率——相当于纵坐标除以横坐标。从图形上看，tan θ 可看作过点 (1,0) 且与单位圆相切的竖直线段被终边所截取的长度。这种几何表示解释了符号规律：tan θ 在第一和第三象限为正（sin 和 cos 同号），在第二和第四象限为负（异号）。
As the angle increases by π radians, the terminal ray rotates half a circle and points in the opposite direction. The slope therefore repeats because the ratio of sin to cos remains unchanged after a half‑turn. This immediately reveals the fundamental period of the tangent function: tan(x + π) = tan x, so the period is π, not 2π like sine and cosine.
当角度增加 π 弧度时，终边旋转半圈并指向相反方向。斜率因此重复，因为经过半圈后 sin 与 cos 的比值保持不变。这立刻揭示了正切函数的基本周期：tan(x + π) = tan x，所以周期是 π，而不是像正弦和余弦那样的 2π。
3. Graph of y = tan x | y = tan x 的图像
rated by vertical asymptotes. It passes through the origin because tan 0 = 0, and it has x‑intercepts at every integer multiple of π: x = kπ, k ∈ ℤ. Between two consecutive asymptotes, the function rises smoothly from –∞ to +∞, crossing the x‑axis exactly halfway. A typical branch between – ated S. The graph is symmetric about the origin, confirming that tan x is an odd function: tan(–x) = –tan x.
y = tan x 的图像由若干条被垂直渐近线隔开的独立且全等的分支组成。它经过原点，因为 tan 0 = 0，并且在 π 的每个整数倍处都有 x 截距：x = kπ, k ∈ ℤ。在两条相邻渐近线之间，函数从 –∞ 平滑地上升到 +∞，并在正中间穿过 x 轴。介于 –π/2 和 π/2 之间的典型分支是严格递增的，形状像一个拉长的 S 形。图像关于原点对称，这证实了 tan x 是奇函数：tan(–x) = –tan x。
There is no maximum or minimum value; the curve never settles to a horizontal level. The steepness of the curve increases dramatically close to the asymptotes, reflecting the unbounded growth of tan x when cos x approaches zero.
图像既没有最大值也没有最小值；曲线永远不会趋近水平。在靠近渐近线时，曲线的陡峭程度急剧增加，反映出当 cos x 趋近于零时 tan x 的无界增长。
4. Asymptotes and Periodicity | 渐近线与周期性
The vertical asymptotes occur exactly where cos x = 0, i.e. at x = π/2 + kπ . Approaching from the left of an asymptote, tan x tends to +∞; from the right it dives to – rmissible x is π. The proof uses the addition formulas: tan(x + π) = sin(x+π)/cos(x+π) = (–sin x)/(–cos x) = tan x. Note that doubling π to 2π also works, but π is the primitive period.
垂直渐近线精确地出现在 cos x = 0 的位置，即 x = π/2 + kπ（k 为任意整数）。从一条渐近线的左侧趋近时，tan x 趋于 +∞；从右侧则陡然降至 –∞。这一模式交替重复，使得每一个分支都呈现相同的行为。满足对所有允许的 x 都有 tan(x + P) = tan x 的最小正周期 P 是 π。证明可以使用加法公式：tan(x + π) = sin(x+π)/cos(x+π) = (–sin x)/(
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