Trigonometric Functions and Their Graph Properties | 三角函数及其图像性质

📚 Trigonometric Functions and Their Graph Properties | 三角函数及其图像性质

Trigonometric functions are one of the most important topics in A-Level mathematics, forming the foundation for calculus, geometry, and many real-world applications. This article provides a comprehensive guide to understanding the six trigonometric functions, their graphs, key properties, and transformations — exactly what you need for exam success.

三角函数是 A-Level 数学中最重要的主题之一，是微积分、几何以及许多实际应用的基础。本文将全面讲解六个三角函数的定义、图像、关键性质以及图像变换——这正是你在考试中取得高分所需要的内容。

1. The Basic Trigonometric Functions | 基本三角函数

The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). For a right-angled triangle with an angle θ, these are defined as ratios of side lengths. The reciprocal functions — cosecant (csc), secant (sec), and cotangent (cot) — are their multiplicative inverses.

三个基本三角函数是正弦（sin）、余弦（cos）和正切（tan）。对于含角 θ 的直角三角形，它们被定义为边长的比值。倒数函数——余割（csc）、正割（sec）和余切（cot）——分别是它们的乘法逆元。

Function	Definition	Reciprocal
sin θ	opposite / hypotenuse	csc θ = 1/sin θ
cos θ	adjacent / hypotenuse	sec θ = 1/cos θ
tan θ	opposite / adjacent	cot θ = 1/tan θ

The tangent function can also be expressed as tan θ = sin θ / cos θ. This relationship is fundamental and appears frequently in trigonometric identities and equations.

正切函数也可以表示为 tan θ = sin θ / cos θ。这一关系是基本的，经常出现在三角恒等式和方程中。

2. Radians and the Unit Circle | 弧度和单位圆

In A-Level mathematics, angles are measured in radians rather than degrees. One full revolution equals 2π radians, so 180° = π radians. The conversion formulas are: degrees × π/180 = radians, and radians × 180/π = degrees.

在 A-Level 数学中，角度以弧度而非度来度量。一整圈等于 2π 弧度，因此 180° = π 弧度。换算公式为：度 × π/180 = 弧度，弧度 × 180/π = 度。

The unit circle is a circle of radius 1 centred at the origin. For any angle θ measured counterclockwise from the positive x-axis, the coordinates of the point where the terminal side intersects the circle are (cos θ, sin θ). This gives a powerful geometric interpretation of trigonometric functions that extends beyond right-angled triangles.

单位圆是以原点为圆心、半径为 1 的圆。对于从正 x 轴逆时针测量的任意角 θ，终边与圆的交点坐标为 (cos θ, sin θ)。这为三角函数提供了强大的几何解释，使其适用范围超越直角三角形。

sin²θ + cos²θ = 1

This fundamental Pythagorean identity follows directly from the unit circle definition and is the basis for many other identities. From it, we also derive: 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.

这个基本的毕达哥拉斯恒等式直接从单位圆定义得出，是许多其他恒等式的基础。由此，我们还推导出：1 + tan²θ = sec²θ 和 1 + cot²θ = csc²θ。

3. Period and Amplitude | 周期和振幅

The sine and cosine functions are periodic with a period of 2π radians. This means their graphs repeat every 2π units. The tangent function has a smaller period of π radians. Periodicity is a defining property of trigonometric functions.

正弦函数和余弦函数是周期函数，周期为 2π 弧度。这意味着它们的图像每 2π 个单位重复一次。正切函数的周期更小，为 π 弧度。周期性是三角函数的定义性特征。

Amplitude is the maximum distance from the equilibrium (middle) position of the graph. For y = sin x and y = cos x, the amplitude is 1, meaning the graphs oscillate between -1 and 1. Amplitude is always a positive value.

振幅是图像距离平衡（中间）位置的最大距离。对于 y = sin x 和 y = cos x，振幅为 1，即图像在 -1 和 1 之间振荡。振幅始终为正值。

Function	Period	Range	Key Points (0 to 2π)
y = sin x	2π	[-1, 1]	0, π/2, π, 3π/2, 2π → 0, 1, 0, -1, 0
y = cos x	2π	[-1, 1]	0, π/2, π, 3π/2, 2π → 1, 0, -1, 0, 1
y = tan x	π	All real numbers	0, π/4, π/2, 3π/4, π → 0, 1, undefined, -1, 0

The sine graph passes through the origin and increases from 0 to 1 at π/2, while the cosine graph starts at its maximum value of 1 at x = 0. The tangent graph has vertical asymptotes at x = π/2 + nπ where the function is undefined.

正弦图像经过原点，从 0 增加到 π/2 处的 1；余弦图像在 x = 0 处从最大值 1 开始。正切图像在 x = π/2 + nπ 处有垂直渐近线，在这些位置函数无定义。

4. Transformations of Trigonometric Graphs | 三角函数图像的变换

Understanding transformations is essential for sketching and interpreting trigonometric graphs. There are four main types of transformations: vertical stretch/compression, horizontal stretch/compression, vertical translation, and horizontal translation.

理解变换对于绘制和解读三角函数图像至关重要。主要有四种变换类型：垂直伸缩、水平伸缩、垂直平移和水平平移。

Vertical stretch/compression: The graph of y = a sin x has amplitude |a|. If a > 1, the graph stretches vertically; if 0 < a < 1, it compresses. If a is negative, the graph is reflected across the x-axis.

垂直伸缩：y = a sin x 的图像振幅为 |a|。若 a > 1，图像垂直拉伸；若 0 < a < 1，则垂直压缩。若 a 为负，图像关于 x 轴反射。

Horizontal stretch/compression: The graph of y = sin(bx) has period 2π/b. If b > 1, the graph compresses horizontally (more oscillations in the same interval); if 0 < b < 1, it stretches. This is a common source of error — students often confuse the directions.

水平伸缩：y = sin(bx) 的图像的周期为 2π/b。若 b > 1，图像水平压缩（同一区间内振荡次数更多）；若 0 < b < 1，则水平拉伸。这是常见的错误来源——学生经常混淆方向。

5. Phase Shift and Vertical Shift | 相位移动和垂直平移

For a function of the form y = a sin(bx + c) + d, the constant c produces a horizontal shift (phase shift), and d produces a vertical shift.

对于形式为 y = a sin(bx + c) + d 的函数，常数 c 产生水平移动（相位移动），而 d 产生垂直平移。

Phase shift: The graph of y = sin(bx + c) is shifted horizontally by -c/b units compared to y = sin(bx). If -c/b > 0, the shift is to the right; if -c/b < 0, the shift is to the left. Always factor out b first before determining the shift direction.

相位移动：y = sin(bx + c) 的图像相对于 y = sin(bx) 水平移动 -c/b 个单位。若 -c/b > 0，则向右移动；若 -c/b < 0，则向左移动。确定移动方向前务必先将 b 因式分解出来。

Vertical shift: The graph of y = sin x + d is shifted upward by d units if d > 0, and downward by |d| units if d < 0. The new equilibrium position becomes y = d, and the new range is [d - |a|, d + |a|].

垂直平移：y = sin x + d 的图像在 d > 0 时上移 d 个单位，在 d < 0 时下移 |d| 个单位。新的平衡位置变为 y = d，新的值域为 [d - |a|, d + |a|]。

y = a sin(bx + c) + d 的通用形式：振幅 = |a|，周期 = 2π/b，相位移动 = -c/b，垂直平移 = d

Let us work through an example. Consider y = 3 cos(2x – π/2) + 1. Here, a = 3, b = 2, c = -π/2, d = 1. The amplitude is 3, the period is 2π/2 = π, the phase shift is -(-π/2)/2 = π/4 (rightward), and the vertical shift is 1 unit upward. The range is [1 – 3, 1 + 3] = [-2, 4].

我们来解一个例子。考虑 y = 3 cos(2x – π/2) + 1。这里，a = 3，b = 2，c = -π/2，d = 1。振幅为 3，周期为 2π/2 = π，相位移动为 -(-π/2)/2 = π/4（向右），垂直平移为向上 1 个单位。值域为 [1 – 3, 1 + 3] = [-2, 4]。

6. Sketching Trigonometric Graphs | 绘制三角函数图像

When sketching trigonometric graphs in exams, it is essential to mark all key features clearly: the equilibrium line, maximum and minimum points, x-intercepts, and asymptotes where applicable.

在考试中绘制三角函数图像时，务必清晰标出所有关键特征：平衡线、最大值和最小值点、x 截距以及渐近线（如适用）。

Step-by-step sketching method:

Identify the amplitude |a|, period 2π/b, phase shift -c/b, and vertical shift d from the equation.
Draw the equilibrium line at y = d.
Mark the maximum and minimum lines at y = d ± |a|.
Determine one full cycle interval: from the phase shift to phase shift + period.
Plot the key points: start, quarter-period, half-period, three-quarter-period, and end.
Connect the points with a smooth curve, respecting the shape of the base function.

从方程中确定振幅 |a|、周期 2π/b、相位移动 -c/b 和垂直平移 d。
在 y = d 处画平衡线。
在 y = d ± |a| 处标出最大值线和最小值线。
确定一个完整周期的区间：从相位移动到相位移动 + 周期。
标出关键点：起点、四分之一周期、半周期、四分之三周期和终点。
用平滑曲线连接各点，保持基函数的形状特征。

For y = tan x, instead of maximum and minimum points, look for vertical asymptotes. The graph approaches these asymptotes but never touches them. The branch between two consecutive asymptotes passes through the centre point where the function value is zero.

对于 y = tan x，不是寻找最大值和最小值点，而是寻找垂直渐近线。图像趋近于这些渐近线但永远不会触及。两条相邻渐近线之间的分支经过函数值为零的中心点。

7. Inverse Trigonometric Functions | 反三角函数

Since trigonometric functions are periodic, they are not one-to-one and therefore do not have true inverses over their entire domains. To define inverse functions, we restrict the domains to intervals where the functions are one-to-one.

由于三角函数是周期函数，它们不是一一对应的，因此在整个定义域上不存在严格意义上的反函数。为了定义反函数，我们需要将定义域限制在函数一一对应的区间上。

Arcsin (sin⁻¹): The inverse of sine, defined on the restricted domain [-π/2, π/2] for the input of sin x, giving outputs in [-1, 1]. The graph of y = arcsin x has range [-π/2, π/2].

Arcsin（sin⁻¹）：正弦函数的反函数，定义域（作为 sin x 的输入）限制在 [-π/2, π/2]，输出范围为 [-1, 1]。y = arcsin x 的图像值域为 [-π/2, π/2]。

Arccos (cos⁻¹): The inverse of cosine, with restricted domain [0, π], giving outputs in [-1, 1], and range [0, π].

Arccos（cos⁻¹）：余弦函数的反函数，定义域限制在 [0, π]，输出范围为 [-1, 1]，值域为 [0, π]。

Arctan (tan⁻¹): The inverse of tangent, with restricted domain (-π/2, π/2), giving outputs in all real numbers, and range (-π/2, π/2). The graph has horizontal asymptotes at y = π/2 and y = -π/2.

Arctan（tan⁻¹）：正切函数的反函数，定义域限制在 (-π/2, π/2)，输出为全体实数，值域为 (-π/2, π/2)。图像在 y = π/2 和 y = -π/2 处有水平渐近线。

Note the notation: sin⁻¹ x does not mean 1/sin x. It represents the inverse function. The reciprocal of sin x is written as (sin x)⁻¹ or csc x to avoid confusion.

注意符号：sin⁻¹ x 并不表示 1/sin x。它代表反函数。sin x 的倒数应写为 (sin x)⁻¹ 或 csc x，以免混淆。

8. Solving Trigonometric Equations | 求解三角方程

Solving trigonometric equations requires finding all values of x that satisfy the equation within a given interval. Because trigonometric functions are periodic, equations typically have multiple solutions.

求解三角方程需要找出在给定区间内满足方程的所有 x 值。由于三角函数具有周期性，方程通常有多个解。

General method:

Rearrange the equation to isolate a single trigonometric function.
Use the inverse function to find the principal value (the solution from the inverse function’s range).
Use the symmetry properties of the trigonometric graphs to find other solutions in the required interval.
Add multiples of the period to generate all possible solutions.

整理方程，分离出单一的三角函数。
使用反函数求主值（反函数值域中的解）。
利用三角函数图像的对称性，在所需区间内找到其他解。
加上周期的整数倍以生成所有可能的解。

For example, solve sin x = 0.5 for 0 ≤ x < 2π. The principal value is arcsin(0.5) = π/6. By the symmetry of the sine graph (sin(π - x) = sin x), the second solution is π - π/6 = 5π/6. The solutions are x = π/6 and x = 5π/6.

例如，在 0 ≤ x < 2π 内求解 sin x = 0.5。主值为 arcsin(0.5) = π/6。根据正弦图像的对称性（sin(π - x) = sin x），第二个解为 π - π/6 = 5π/6。解为 x = π/6 和 x = 5π/6。

For cosine, the symmetry is cos(2π – x) = cos x. For tangent, the symmetry is tan(x + π) = tan x, meaning solutions repeat every π units. Knowing these symmetries allows you to find all solutions systematically.

对于余弦，对称性为 cos(2π – x) = cos x。对于正切，对称性为 tan(x + π) = tan x，意味着解每 π 个单位重复一次。掌握这些对称性可以让你系统地找到所有解。

9. Trigonometric Identities and Proofs | 三角恒等式与证明

Trigonometric identities are equations that hold true for all values of the variable. The most important ones for A-Level mathematics include the Pythagorean identities, double-angle formulas, and compound-angle formulas.

三角恒等式是对变量的所有值都成立的等式。A-Level 数学中最重要的包括毕达哥拉斯恒等式、倍角公式和复合角公式。

Double-angle formulas:

倍角公式：

sin 2θ = 2 sin θ cos θ

cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ

tan 2θ = 2 tan θ / (1 – tan²θ)

Compound-angle formulas:

复合角公式：

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

These identities are frequently tested in both pure mathematics and in applications such as solving equations. When proving identities, the general strategy is to start from the more complex side and simplify it step by step until it matches the simpler side, using known identities at each stage.

这些恒等式在纯数学和方程求解等应用中经常被考查。证明恒等式时，一般策略是从较复杂的一边入手，利用已知恒等式逐步化简，直到与较简单的一边一致。

10. Applications and Exam Tips | 应用与考试技巧

Trigonometric graphs and functions appear throughout A-Level mathematics, including in calculus (differentiating and integrating trig functions), mechanics (simple harmonic motion), and solving real-world problems involving periodic phenomena.

三角函数图像和函数贯穿 A-Level 数学的各个部分，包括微积分（三角函数的微分和积分）、力学（简谐运动）以及涉及周期性现象的实际问题。

Common exam mistakes to avoid:

需要避免的常见考试错误：

Using degrees instead of radians when the question specifies radians — always check the units specified.
Forgetting to factor out b before finding the phase shift of a sin(bx + c).
Confusing the horizontal stretch: y = sin(2x) has period π, NOT 4π.
Missing solutions when solving trigonometric equations — remember the periodicity.
Writing sin⁻¹ x to mean 1/sin x — this is incorrect notation.
Forgetting that the range of sin x and cos x is [-1, 1], so equations like sin x = 2 have no solutions.

当题目指定弧度时仍使用度数——始终检查题目指定的单位。
在求 sin(bx + c) 的相位移动前忘记先因式分解出 b。
混淆水平伸缩：y = sin(2x) 的周期是 π，而不是 4π。
求解三角方程时遗漏解——切记周期性。
将 sin⁻¹ x 写成表示 1/sin x——这是错误符号。
忘记 sin x 和 cos x 的值域是 [-1, 1]，因此 sin x = 2 这样的方程无解。

Key facts to memorise for exams:

考试需牢记的关键事实：

sin x and cos x have period 2π; tan x has period π.
sin x is an odd function (sin(-x) = -sin x); cos x is an even function (cos(-x) = cos x).
The graphs of sin x and cos x are identical except for a horizontal shift of π/2.
The gradient of y = sin x at x = 0 is 1, which connects to the small-angle approximation sin x ≈ x for small x.

sin x 和 cos x 的周期为 2π；tan x 的周期为 π。
sin x 是奇函数（sin(-x) = -sin x）；cos x 是偶函数（cos(-x) = cos x）。
sin x 和 cos x 的图像除了相差 π/2 的水平移动外完全相同。
y = sin x 在 x = 0 处的斜率为 1，这与小角度近似 sin x ≈ x（x 很小时）相关联。

Finally, always sketch a quick graph when solving trigonometric problems — even a rough sketch can help you identify how many solutions to expect and verify the reasonableness of your answers. Practice regularly with past paper questions to build fluency with graph transformations and equation solving.

最后，解决三角问题时始终快速画一张草图——即使是粗略的草图也能帮助你判断预期的解的数量，并验证答案的合理性。定期练习历年真题，以熟练掌握图像变换和方程求解。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Trigonometric Functions and Their Graph Properties | 三角函数及其图像性质