Graphs and Properties of Trigonometric Functions | 三角函数的图像与性质

📚 Graphs and Properties of Trigonometric Functions | 三角函数的图像与性质

The trigonometric functions sine, cosine, and tangent form the backbone of A-Level mathematics. Mastering their graphs and properties is essential for solving equations, modelling periodic phenomena, and understanding transformations. This guide covers every examinable detail in a clear, structured way.

正弦、余弦和正切是 A-Level 数学的核心内容。掌握它们的图像与性质，是解三角方程、建立周期现象模型以及理解图像变换的关键。本文将用清晰、结构化的方式，逐一梳理所有考试要点。

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1. The Sine Function y = sin x | 正弦函数 y = sin x

s a minimum of -1 at x = 3π/2, and returns to 0 at x = 2π. The wave then repeats every 2π.

y = sin x 的图像是一条平滑的波浪线，经过原点 (0, 0)。它在 x = π/2 处上升到最大值 1，在 x = π 处回到 0，在 x = 3π/2 处达到最小值 -1，在 x = 2π 处回到 0。此后波形每 2π 重复一次。

the period (length of one complete cycle) is 2π. The midline is the horizontal line y = 0.

振幅（偏离中轴线的最大距离）为 1，周期（一个完整循环的长度）为 2π。中轴线为水平直线 y = 0。

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2. The Cosine Function y = cos x | 余弦函数 y = cos x

at x = π/2, reaches -1 at x = π, returns to 0 at x = 3π/2, and rises back to 1 at x = 2π.

y = cos x 的图像与 y = sin x 形状相同，但起点在最大值处：(0, 1)。它在 x = π/2 处降到 0，在 x = π 处达到 -1，在 x = 3π/2 处回到 0，并在 x = 2π 处回升到 1。

The cosine graph is simply the sine graph shifted left by π/2. This gives the fundamental identity:

余弦图像就是正弦图像向左平移 π/2 的结果。由此得到基本恒等式：

cos x = sin(x + π/2)

Both sine and cosine have amplitude 1 and period 2π, but they differ in their starting positions.

正弦和余弦的振幅均为 1，周期均为 2π，区别仅在于起点位置不同。

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3. The Tangent Function y = tan x | 正切函数 y = tan x

otes at x = π/2 + nπ (where n is any integer), because tan x is undefined where cos x = 0.

y = tan x 的图像与正弦、余弦截然不同。它经过原点且斜率陡峭，但在 x = π/2 + nπ（n 为任意整数）处有垂直渐近线，因为在这些位置 cos x = 0，tan x 无定义。

rises from -∞ to +∞, crossing the x-axis at x = nπ. The period of y = tan x is π, not 2π.

在相邻渐近线之间，图像从 -∞ 上升到 +∞，并在 x = nπ 处穿过 x 轴。y = tan x 的周期是 π，而不是 2π。

tan x = sin x / cos x, x ≠ π/2 + nπ

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4. Domain and Range | 定义域与值域

For A-Level exams, you must state the domain and range of each trigonometric function precisely.

在 A-Level 考试中，你必须准确写出每个三角函数的定义域和值域。

The sine and cosine functions are defined for all real numbers. Their outputs are always between -1 and 1 inclusive:

正弦和余弦函数对所有实数都有定义。它们的输出值始终在 -1 到 1 之间（含端点）：

• Domain of sin x and cos x: all real numbers, x ∈ ℝ

  sin x 和 cos x 的定义域：一切实数，x ∈ ℝ

• Range of sin x and cos x: -1 ≤ y ≤ 1

  sin x 和 cos x 的值域：-1 ≤ y ≤ 1

ore its domain excludes all odd multiples of π/2. Its range, however, is all real numbers:

正切函数在 cos x = 0 处无定义，因此其定义域排除所有 π/2 的奇数倍。但其值域为一切实数：

• Domain of tan x: x ≠ π/2 + nπ, where n ∈ ℤ

  tan x 的定义域：x ≠ π/2 + nπ，其中 n ∈ ℤ

• Range of tan x: y ∈ ℝ (all real numbers)

  tan x 的值域：y ∈ ℝ（一切实数）

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5. Period and Amplitude | 周期与振幅

The period of a periodic function is the length of the smallest interval over which the function repeats. The amplitude is half the vertical distance between the maximum and minimum values.

周期函数的最小重复区间长度称为周期。振幅是最大值与最小值之间垂直距离的一半。

For the standard functions:

对于标准函数：

Function	Period	Amplitude
y = sin x	2π	1
y = cos x	2π	1
y = tan x	π	Not defined

function of the form y = a sin(bx) or y = a cos(bx), where a and b are positive constants:

对于形如 y = a sin(bx) 或 y = a cos(bx) 的函数，其中 a、b 为正常数：

Period = 2π / b, Amplitude = |a|

For y = a tan(bx), the period is π / b.

对于 y = a tan(bx)，周期为 π / b。

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6. Transformations: y = a sin(bx + c) + d | 图像变换

In A-Level exams, you are often asked to sketch or describe transformations of trigonometric graphs. The general form combines four types of change:

在 A-Level 考试中，经常要求你画出或描述三角函数图像的变换。一般形式包含四种类型的变化：

y = a sin(bx + c) + d

• The parameter a changes the amplitude. If a is negative, the graph is also reflected in the x-axis.

  参数 a 改变振幅。若 a 为负数，图像还会关于 x 轴反射。

• The parameter b changes the period. The new period is 2π / b. Larger b compresses the graph horizontally.

  参数 b 改变周期。新周期为 2π / b。b 越大，图像在水平方向被压缩得越紧。

• The parameter d shifts the graph vertically. The midline becomes y = d, with maximum d + |a| and minimum d - |a|.

  参数 d 使图像垂直平移。中轴线变为 y = d，最大值为 d + |a|，最小值为 d - |a|。

These transformations apply identically to cosine graphs, and the vertical stretch and shift also apply to tangent graphs (though tangent has no amplitude).

这些变换同样适用于余弦图像，垂直拉伸和平移也适用于正切图像（但正切没有振幅）。

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7. Phase Shift and Angle c | 相位移与角 c

nslation called the phase shift. The graph shifts left or right relative to y = a sin(bx).

在 y = a sin(bx + c) 中，参数 c 产生水平平移，称为相位移。图像相对于 y = a sin(bx) 向左或向右移动。

The phase shift is calculated as:

相位移的计算方式为：

Phase shift = -c / b

π/3) has a phase shift of π/3 to the right, while y = sin(x + π/3) shifts π/3 to the left.

若 -c/b 为正，图像向右平移；若为负，则向左平移。例如 y = sin(x - π/3) 向右平移 π/3，而 y = sin(x + π/3) 向左平移 π/3。

For y = sin(2x + π), rewrite as y = sin(2(x + π/2)) to see that the shift is -π/2, not -π.

注意：当系数 b 不为 1 时，必须先提取公因子。对于 y = sin(2x + π)，改写为 y = sin(2(x + π/2)) 后可知相位移是 -π/2，而不是 -π。

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8. Symmetry and Parity | 对称性与奇偶性

Trigonometric functions have important symmetry properties that simplify calculations and graphing.

三角函数具有重要的对称性质，可简化计算和作图。

The sine function is odd. This means:

正弦函数是奇函数，即：

sin(-x) = -sin x

Its graph is symmetric about the origin: rotating it 180° about (0, 0) leaves it unchanged.

其图像关于原点对称：绕 (0, 0) 旋转 180° 后与自身重合。

The cosine function is even. This means:

余弦函数是偶函数，即：

cos(-x) = cos x

Its graph is symmetric about the y-axis.

其图像关于 y 轴对称。

The tangent function is also odd:

正切函数同样是奇函数：

tan(-x) = -tan x

These identities are frequently used in exam questions to evaluate functions at negative angles or to simplify expressions.

这些恒等式在考试中常用于计算负角度的函数值或化简表达式。

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9. Key Points and Special Angles | 关键点与特殊角

You should memorise the exact values of sine, cosine, and tangent at the special angles, as they appear in nearly every exam paper.

你必须牢记正弦、余弦和正切在特殊角处的精确值，因为它们在几乎每张试卷中都会出现。

x	0	π/6	π/4	π/3	π/2
sin x	0	1/2	√2/2	√3/2	1
cos x	1	√3/2	√2/2	1/2	0
tan x	0	1/√3	1	√3	Undefined

n x, the five-point pattern over one period is (0, 0), (π/2, 1), (π, 0), (3π/2, -1), (2π, 0).

画图时，标出关键点：截距、最大值点、最小值点和渐近线。对于 y = sin x，一个周期内的五点模式为 (0, 0)、(π/2, 1)、(π, 0)、(3π/2, -1)、(2π, 0)。

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10. Solving Trigonometric Equations Graphically | 图像法解三角方程

a given interval. The graph method is reliable: draw the curve and a horizontal line, then read off the intersections.

常见考题要求找出 sin x = 0.5 这类方程在给定区间内的解个数或精确解。图像法最为可靠：画出曲线和水平直线，然后读出交点。

For sin x = 0.5 with 0 ≤ x ≤ 2π, the sine graph intersects the line y = 0.5 twice:

对于 sin x = 0.5，在 0 ≤ x ≤ 2π 上，正弦图像与直线 y = 0.5 相交两次：

x = π/6 and x = 5π/6

unless the value is ±1), while tangent equations have exactly one solution in each period.

一般规律是：正弦和余弦方程在每个周期内通常有两个解（除非取值为 ±1），而正切方程在每个周期内恰好有一个解。

ans. If the interval is 0° ≤ x ≤ 360°, use degree mode; if it involves π, use radian mode.

始终检查区间是用度还是弧度给出的。若区间为 0° ≤ x ≤ 360°，使用角度模式；若涉及 π，则使用弧度模式。

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11. Common Exam Pitfalls | 常见考试易错点

Even strong students lose marks on trigonometric graphs due to avoidable mistakes. Watch out for these traps.

即使是优秀学生，也常在三角函数图像上因可避免的错误而失分。请务必小心以下陷阱。

• Forgetting to set the calculator to the correct mode. Mixing degrees and radians leads to completely wrong answers.

  忘记将计算器设置为正确的角度模式。混用度与弧度会导致答案完全错误。

• Writing the period of tan x as 2π. The correct period is π.

  把 tan x 的周期写成 2π。正确周期是 π。

• In y = sin(x + c), the shift is -c. A plus sign means a shift to the left, not the right.

  搞错相位移的符号。对于 y = sin(x + c)，位移为 -c。加号表示向左平移，而不是向右。

• Forgetting vertical asymptotes when sketching y = tan x or y = a tan(bx).

  画 y = tan x 或 y = a tan(bx) 时忘记标出垂直渐近线。

• ing equations. For example, sin x = 1/2 in 0 ≤ x ≤ 2π has two solutions, not just x = π/6.

  解方程时漏解。例如在 0 ≤ x ≤ 2π 内，sin x = 1/2 有两个解，而不仅仅是 x = π/6。

• e of a transformed function. For y = 3 sin x + 2, the range is -1 ≤ y ≤ 5, not -3 ≤ y ≤ 3.

  求变换后函数的值域时忽略中轴线位移 d。对于 y = 3 sin x + 2，值域是 -1 ≤ y ≤ 5，而不是 -3 ≤ y ≤ 3。

Reviewing these pitfalls before your exam will save you from classic errors and boost your confidence.

考前复习这些易错点，能帮助你避开常见错误，增强信心。

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Understanding the graphs of sine, cosine, and tangent is not just about memorising shapes. It requires connecting amplitude, period, phase shift, symmetry, and key values into one coherent picture. Practise sketching each graph from memory, and always check your transformed graphs against the original features.

理解正弦、余弦和正切的图像，不仅仅是记住形状。你需要将振幅、周期、相位移、对称性和关键值整合成一幅连贯的图像。练习凭记忆画出每个函数图像，并始终用原始特征检验变换后的图像。

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