📚 Graphs of Trigonometric Functions | 三角函数图像
Trigonometric functions model periodic phenomena from sound waves to planetary motion. Understanding their graphs is essential in IB Mathematics, both for visualising solutions and for applying transformations. This article explores the core graphs of sine, cosine and tangent, their reciprocal functions, and how parameters change their shape and position.
三角函数可用于模拟从声波到行星运动的周期现象。掌握它们的图像在 IB 数学中至关重要，既能直观地理解解的情况，也能进行图像变换。本文将探讨正弦、余弦和正切的基本图像、它们的倒数函数以及参数如何改变图像的形状和位置。
1. The Basic Sine Curve y = sin x | 基础正弦曲线 y = sin x
The graph of y = sin x is a smooth wave that oscillates between −1 and 1. It starts at the origin (0,0), reaches a maximum of 1 at x = π/2, returns to 0 at x = π, reaches a minimum of −1 at x = 3π/2, and completes one full cycle at x = 2π.
y = sin x 的图像是一条在 −1 和 1 之间振荡的平滑波形。它从原点 (0,0) 出发，在 x = π/2 处达到最大值 1，在 x = π 处回到 0，在 x = 3π/2 处达到最小值 −1，并在 x = 2π 处完成一个完整周期。
The period of sin x is 2π, meaning the pattern repeats every 2π radians. Its amplitude is 1. The function is odd, so its graph is symmetric about the origin.
sin x 的周期为 2π，意味着每 2π 弧度波形重复一次。振幅为 1。该函数是奇函数，因此图像关于原点对称。
2. The Cosine Function y = cos x | 余弦函数 y = cos x
The graph of y = cos x is also a wave of amplitude 1 and period 2π, but it starts at its maximum value of 1 when x = 0. It crosses the x-axis at π/2 and 3π/2, reaches a minimum of −1 at x = π, and completes a cycle at 2π.
y = cos x 的图像同样是振幅为 1、周期为 2π 的波形，但在 x = 0 时从最大值 1 开始。它在 π/2 和 3π/2 处穿过 x 轴，在 x = π 处达到最小值 −1，并在 2π 处完成一个周期。
Cosine is an even function; its graph is symmetric about the y-axis. In fact, the cosine curve is a horizontally shifted sine curve: cos x = sin(x + π/2).
余弦是偶函数，其图像关于 y 轴对称。实际上，余弦曲线是正弦曲线水平平移的结果：cos x = sin(x + π/2)。
3. The Tangent Function y = tan x | 正切函数 y = tan x
The graph of y = tan x looks completely different from the sine and cosine waves. It consists of repeated branches separated by vertical asymptotes at x = π/2 + kπ, where k is an integer. The period of tan x is π, half the period of sine and cosine.
y = tan x 的图像与正弦、余弦波完全不同。它由重复的分支组成，分支之间由 x = π/2 + kπ（k 为整数）处的垂直渐近线分隔。tan x 的周期为 π，是正弦和余弦周期的一半。
The curve passes through the origin, is increasing on each interval between asymptotes, and has no maximum or minimum values – its range is all real numbers. The function is odd, symmetric about the origin.
曲线经过原点，在每条渐近线之间的区间上单调递增，且无最大值或最小值——值域为全体实数。该函数为奇函数，关于原点对称。
4. Amplitude, Period and Phase Shift | 振幅、周期和相位移动
For a transformed sine or cosine function written as y = A sin(B(x − C)) + D, the parameters have clear meanings. The amplitude is |A|, the vertical stretch factor. The period is given by 2π/|B|. The phase shift (horizontal translation) is C: if C > 0 the graph shifts to the right, if C < 0 it shifts to the left.
对于变换后的正弦或余弦函数 y = A sin(B(x − C)) + D，每个参数都有明确的含义。振幅为 |A|，即纵向伸缩因子。周期为 2π/|B|。相位移动（水平平移）为 C：若 C > 0，图像向右平移；若 C < 0，图像向左平移。
For example, y = 3 sin(2(x − π/4)) has amplitude 3, period π, and a phase shift of π/4 to the right. Always rewrite the function in the form B(x − C) to identify C correctly.
例如，y = 3 sin(2(x − π/4)) 的振幅为 3，周期为 π，且向右平移 π/4。始终将函数改写为 B(x − C) 的形式，以便正确确定 C。
5. Vertical Translations | 垂直平移
The parameter D in y = A sin(B(x − C)) + D shifts the entire graph vertically. The midline, which is normally y = 0, moves to y = D. Consequently, the maximum and minimum values become D + |A| and D − |A| respectively.
y = A sin(B(x − C)) + D 中的参数 D 可使整个图像上下移动。原本位于 y = 0 的中线移至 y = D。因此，最大值和最小值分别变为 D + |A| 和 D − |A|。
If D is positive, the graph shifts upward; if negative, downward. This is often combined with amplitude and period changes to model real-world data, such as average temperature variations.
若 D 为正，图像上移；为负则下移。这常与振幅、周期变化结合，用于拟合现实数据，如平均气温变化。
6. Sketching Transformed Graphs | 绘制变换图像
To sketch y = A sin(B(x − C)) + D accurately, start with the basic sin x curve. Apply horizontal scaling by 1/B, then shift horizontally by C, stretch vertically by |A|, and finally translate vertically by D. Work in this order to avoid mistakes.
要准确绘制 y = A sin(B(x − C)) + D，可先从基础的 sin x 曲线入手。先进行水平伸缩（因子 1/B），再水平平移 C，然后纵向拉伸 |A|，最后纵向平移 D。按此顺序操作可避免错误。
Key points to plot are the starting point (C, D), maxima, minima, and intercepts with the midline. Always label the period and amplitude on your sketch.
需要标出的关键点包括起点 (C, D)、最大值点、最小值点以及与中线的交点。绘制草图时应始终标注周期和振幅。
7. Reciprocal Functions: Cosecant, Secant and Cotangent | 倒数函数：余割、正割、余切
The reciprocal functions are defined as csc x = 1/sin x, sec x = 1/cos x, and cot x = 1/tan x. Their graphs are closely related to the original functions but feature vertical asymptotes where the denominator is zero.
倒数函数定义为 csc x = 1/sin x，sec x = 1/cos x，cot x = 1/tan x。它们的图像与原函数密切相关，但在分母为零处有垂直渐近线。
For y = csc x, asymptotes occur at x = kπ; the graph consists of U-shaped branches opening upward from y = 1 and downward from y = −1. For y = sec x, asymptotes are at x = π/2 + kπ, with branches opening upward from y = 1 and downward from y = −1. The cotangent graph has asymptotes at x = kπ and decreases on each interval.
y = csc x 的渐近线位于 x = kπ；图像由开口向上（从 y = 1 开始）和开口向下（从 y = −1 开始）的 U 型分支组成。y = sec x 的渐近线在 x = π/2 + kπ 处，分支分别从 y = 1 向上和从 y = −1 向下延伸。余切图像在 x = kπ 处有渐近线，并在每个区间内单调递减。
8. Asymptotes and Domain Restrictions | 渐近线与定义域限制
Vertical asymptotes play a crucial role in trigonometric graphs. For y = tan x, they appear at x = π/2 + kπ; for y = cot x, at x = kπ. The reciprocal functions inherit asymptotes from the zeros of sine and cosine.
垂直渐近线在三角函数图像中起着关键作用。y = tan x 的渐近线出现在 x = π/2 + kπ；y = cot x 在 x = kπ。倒数函数从正弦、余弦的零点处继承了渐近线。
Always state the domain explicitly. For instance, the domain of y = tan x is all real numbers except x ≠ π/2 + kπ. The domain of y = sec x excludes x = π/2 + kπ, while y = csc x excludes x = kπ.
必须明确写出定义域。例如，y = tan x 的定义域为所有实数，但 x ≠ π/2 + kπ。y = sec x 的定义域排除 x = π/2 + kπ，而 y = csc x 排除 x = kπ。
9. Using the Unit Circle to Visualise Graphs | 利用单位圆可视化图像
The unit circle provides a powerful way to understand why trigonometric graphs have their shapes. As an angle θ increases from 0, the y-coordinate of the point on the circle traces the sine curve, while the x-coordinate traces the cosine curve.
单位圆是理解三角函数图像形状的强大工具。随着角度 θ 从 0 开始增加，圆周上点的 y 坐标描绘出正弦曲线，而 x 坐标描绘出余弦曲线。
For tan θ, think of the slope of the line from the origin to the point on the circle. This slope grows without bound as θ approaches π/2, creating the vertical asymptote. Wrapping the circle onto an axis generates the periodic waves.
对于 tan θ，可将其视为从原点到单位圆上点的直线斜率。当 θ 趋向 π/2 时，斜率无限增大，从而产生垂直渐近线。将圆周“展开”到坐标轴上便生成了周期波形。
10. Solving Trigonometric Equations Graphically | 图像法解三角方程
Equations such as sin x = 0.5 can be solved by graphing y = sin x and y = 0.5 on the same axes. The x-values of the intersection points within the given interval are the solutions. This visual method helps identify multiple solutions and their symmetry patterns.
求解 sin x = 0.5 这类方程时，可在同一坐标系中绘制 y = sin x 和 y = 0.5 的图像。给定区间内交点的 x 坐标即为方程的解。这一图象法有助于识别多个解及其对称规律。
You can also transform equations into the form f(x) = g(x) and graph both sides. The graphical approach is especially useful when dealing with non-standard equations or when checking analytical solutions.
也可以将方程改写为 f(x) = g(x) 的形式，并绘制两边图像。在处理非标准方程或检验解析解时，图像法尤为有用。
11. Summary and Key Tips | 总结与关键提示
To master trigonometric graphs: memorise the basic shapes and key points of sin, cos and tan. Understand the effects of A, B, C and D in the general form. When sketching, always determine the period, amplitude, phase shift and vertical shift first, then plot one cycle and extend it.
要掌握三角函数图像：牢记 sin、cos 和 tan 的基本形状及关键点；理解一般形式中 A、B、C、D 的作用；绘制草图时，务必先确定周期、振幅、相位移动和垂直平移，然后画出一个周期并延伸。
Work exclusively in radians unless the question specifies degrees. Pay attention to asymptotes and domain restrictions for tan and reciprocal functions. Practice transforming between algebraic and graphical representations to build confidence for IB exam questions.
除非题目特别说明，应全程使用弧度制。注意 tan 及倒数函数的渐近线和定义域限制。多加练习代数形式与图像表征之间的转换，以增强应对 IB 考试题目的信心。
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