A-Level Maths: Graphs of Sine, Cosine and Tangent | A-Level数学：正弦、余弦和正切图像

📚 A-Level Maths: Graphs of Sine, Cosine and Tangent | A-Level数学：正弦、余弦和正切图像

The trigonometric functions sine, cosine and tangent are fundamental to A-Level Mathematics. Their graphs appear repeatedly in calculus, solving equations, wave modelling and even mechanics problems. Mastering the shape, key features and transformations of these curves is essential for Edexcel exam success.

正弦、余弦和正切这三个三角函数是A-Level数学的基础。它们的图像在微积分、解方程、波动建模乃至力学问题中反复出现。掌握这些曲线的形状、关键特征及变换规律，是Edexcel考试取得高分的关键。

1. The Sine Graph | 正弦函数图像

The graph of y = sin x is a smooth, continuous wave that repeats every 360° (or 2π radians). It passes through the origin, rises to a maximum of 1 at 90° (π/2), returns to 0 at 180° (π), falls to a minimum of -1 at 270° (3π/2), and completes one full cycle at 360° (2π).

y = sin x 的图像是一条平滑连续的波浪线，每 360°（或 2π 弧度）重复一次。它经过原点，在 90°（π/2）处上升到最大值 1，在 180°（π）处回到 0，在 270°（3π/2）处下降到最小值 -1，并在 360°（2π）处完成一个完整周期。

Range: -1 ≤ sin x ≤ 1 | 值域：-1 ≤ sin x ≤ 1
Period: 360° or 2π radians | 周期：360° 或 2π 弧度
x-intercepts: x = 0, ±π, ±2π, … (x = kπ, k ∈ ℤ) | 与 x 轴交点：x = 0, ±π, ±2π, … (x = kπ, k ∈ ℤ)
Maximum: (π/2 + 2kπ, 1) | 最大值点：(π/2 + 2kπ, 1)
Minimum: (-π/2 + 2kπ, -1) | 最小值点：(-π/2 + 2kπ, -1)

Notice the sine graph is an odd function: sin(-x) = -sin x, meaning it has rotational symmetry about the origin. This is useful for quickly sketching graphs for negative angles.

注意正弦函数是奇函数：sin(-x) = -sin x，即它关于原点具有旋转对称性。这有助于快速画出负角度区域的图像。

2. The Cosine Graph | 余弦函数图像

The graph of y = cos x has the same wave shape as sine but is shifted left by 90°. It starts at its maximum value of 1 when x = 0, falls to 0 at 90°, reaches its minimum -1 at 180°, returns to 0 at 270°, and completes the cycle at 360°.

y = cos x 的图像与正弦函数有相同的波浪形状，但向左平移了 90°。它在 x = 0 时从最大值 1 出发，在 90° 处下降到 0，在 180° 处到达最小值 -1，在 270° 处回到 0，并在 360° 处完成一个周期。

Range: -1 ≤ cos x ≤ 1 | 值域：-1 ≤ cos x ≤ 1
Period: 360° or 2π radians | 周期：360° 或 2π 弧度
x-intercepts: x = π/2 + kπ (k ∈ ℤ) | 与 x 轴交点：x = π/2 + kπ (k ∈ ℤ)
Maximum: (2kπ, 1) | 最大值点：(2kπ, 1)
Minimum: (π + 2kπ, -1) | 最小值点：(π + 2kπ, -1)

The cosine graph is an even function: cos(-x) = cos x, so it is symmetric about the y-axis. Understanding this symmetry helps when solving equations over symmetric intervals such as [-π, π].

余弦函数是偶函数：cos(-x) = cos x，因此它关于 y 轴对称。理解这种对称性有助于在对称区间（如 [-π, π]）内解方程。

3. The Tangent Graph | 正切函数图像

The graph of y = tan x is fundamentally different from sine and cosine. It consists of repeated branches separated by vertical asymptotes, and it has no maximum or minimum values. The tangent function increases from -∞ to +∞ on each open interval between consecutive asymptotes.

y = tan x 的图像与正弦、余弦截然不同。它由被竖直渐近线分隔的重复分支组成，没有最大值或最小值。在相邻两条渐近线之间的每个开区间内，正切函数从 -∞ 递增到 +∞。

Range: all real numbers | 值域：全体实数
Period: 180° or π radians | 周期：180° 或 π 弧度
Vertical asymptotes: x = π/2 + kπ (k ∈ ℤ) | 竖直渐近线：x = π/2 + kπ (k ∈ ℤ)
x-intercepts: x = kπ (k ∈ ℤ) | 与 x 轴交点：x = kπ (k ∈ ℤ)
Key points: tan 0 = 0, tan(π/4) = 1, tan(-π/4) = -1 | 特殊点：tan 0 = 0，tan(π/4) = 1，tan(-π/4) = -1

As x approaches an asymptote from the left, tan x tends to +∞; from the right, it tends to -∞. When sketching, always draw dashed vertical lines at the asymptotes first, then draw the increasing curve through the intercepts.

当 x 从左侧接近渐近线时，tan x 趋于 +∞；从右侧接近时，tan x 趋于 -∞。画图时应先画出渐近线的虚线，再通过 x 轴交点画出递增的曲线分支。

4. Amplitude, Period and Frequency | 振幅、周期和频率

For a general sine or cosine curve y = a sin(bx) + d or y = a cos(bx) + d, the value |a| is called the amplitude. It measures the vertical distance from the central line (y = d) to the maximum or minimum of the wave.

对于一般的正弦或余弦曲线 y = a sin(bx) + d 或 y = a cos(bx) + d，|a| 称为振幅。它表示从中心线（y = d）到波峰或波谷的竖直距离。

The period of y = sin(bx) or y = cos(bx) is 360°/b (in degrees) or 2π/b (in radians). For y = tan(bx), the period is 180°/b or π/b. The frequency is the reciprocal of the period and describes how many cycles occur per unit.

y = sin(bx) 或 y = cos(bx) 的周期是 360°/b（角度制）或 2π/b（弧度制）。对于 y = tan(bx)，周期是 180°/b 或 π/b。频率是周期的倒数，表示单位时间内完成的周期数。

Period of sin/cos = 2π/b | tan = π/b

For example, y = 3 sin(2x) has amplitude 3 and period π. This means the wave oscillates twice as fast as y = sin x, reaching 3 and -3.

例如，y = 3 sin(2x) 的振幅为 3，周期为 π。这意味着该波的振荡速度是 y = sin x 的两倍，最大、最小值分别为 3 和 -3。

5. Phase Shift and Vertical Translation | 相位平移与竖直平移

The graph of y = sin(x – c) is shifted to the right by c units, while y = sin(x + c) is shifted to the left by c units. This horizontal shift is called the phase shift. The value d in y = a sin(x) + d shifts the entire graph vertically upward (d > 0) or downward (d < 0).

y = sin(x – c) 的图像向右平移 c 个单位，而 y = sin(x + c) 的图像向左平移 c 个单位。这种水平平移称为相位平移。y = a sin(x) + d 中的 d 将整条曲线向上（d > 0）或向下（d < 0）平移。

Combining transformations, the general form is y = a sin(b(x – c)) + d. The new central line is y = d, the amplitude is |a|, and the phase shift is c units to the right. Remember to factor out b before reading the phase shift: y = sin(2x – π) = sin(2(x – π/2)), so the shift is π/2, not π.

综合变换时，一般形式为 y = a sin(b(x – c)) + d。新的中心线为 y = d，振幅为 |a|，相位平移为向右 c 个单位。注意：在读取相位平移前需先提取 b：y = sin(2x – π) = sin(2(x – π/2))，因此平移量是 π/2，而不是 π。

y = a sin(b(x – c)) + d → amplitude = |a|, period = 2π/b, shift right = c, vertical shift = d

6. Reflections of Trig Graphs | 三角图像的反射变换

Multiplying by -1 reflects the graph in the x-axis: y = -sin x turns every maximum into a minimum and vice versa. Replacing x by -x reflects the graph in the y-axis: y = sin(-x) = -sin x, which is the reflection of sine in the horizontal axis.

乘以 -1 会使图像关于 x 轴反射：y = -sin x 将每个最大值变为最小值，反之亦然。将 x 替换为 -x 可使图像关于 y 轴反射：y = sin(-x) = -sin x，即正弦函数关于水平轴的镜像。

For cosine, y = cos(-x) = cos x because cosine is even, so no visible change occurs. For tangent, y = tan(-x) = -tan x, so the branches are reflected across the x-axis. When combining reflections with stretches, always sketch the original graph first, then apply transformations in order: stretch, reflect, translate.

对于余弦，y = cos(-x) = cos x，因为余弦是偶函数，所以图像外观不变。对于正切，y = tan(-x) = -tan x，因此各分支关于 x 轴反射。当反射与伸缩同时出现时，先画原始图像，再按顺序进行变换：先伸缩、再反射、最后平移。

y = -f(x): reflect in x-axis | 关于 x 轴反射
y = f(-x): reflect in y-axis | 关于 y 轴反射
y = |f(x)|: reflect the part below x-axis upward | 将 x 轴下方的部分向上翻折

7. Solving Trigonometric Equations Graphically | 利用图像解三角方程

To solve an equation such as sin x = 0.5 for 0 ≤ x ≤ 360°, draw the horizontal line y = 0.5 on the sine graph. The x-coordinates of the intersection points are the solutions: x = 30° and x = 150° in this example.

解形如 sin x = 0.5（0 ≤ x ≤ 360°）的方程时，在正弦图像上画水平线 y = 0.5。交点处的 x 坐标就是解：本例中 x = 30° 和 x = 150°。

For equations involving transformed functions, such as 2 cos(3x – 60°) = -1, rewrite as cos(3x – 60°) = -0.5, sketch the cosine curve (with period 120° and phase shift 20°) and find all intersections in the given interval. Always check how many full cycles fit inside the domain.

对于涉及变换函数的方程，如 2 cos(3x – 60°) = -1，先改写为 cos(3x – 60°) = -0.5，画出余弦曲线（周期为 120°，相位平移 20°），再在给定区间内找出所有交点。务必检查定义域内包含多少个完整周期。

Graphical methods are especially useful when an equation cannot be solved algebraically, such as x + 2 sin x = 1. In such cases, sketch y = x and y = 1 – 2 sin x separately and estimate the intersection point.

图像法在方程无法代数求解时特别有用，例如 x + 2 sin x = 1。此时可分别画出 y = x 和 y = 1 – 2 sin x，并估计交点位置。

8. Symmetry and Trigonometric Identities | 对称性与三角恒等式

The graphs exhibit useful symmetry patterns that aid in solving equations. For sine, sin(π – θ) = sin θ, so a solution at θ implies another at π – θ. For cosine, cos(2π – θ) = cos θ, so a solution at θ implies another at 2π – θ.

图像的对称性有助于解方程。对正弦，sin(π – θ) = sin θ，即若 θ 是一个解，则 π – θ 也是解。对余弦，cos(2π – θ) = cos θ，即若 θ 是一个解，则 2π – θ 也是解。

For tangent, tan(θ + π) = tan θ, so solutions repeat every π radians. These identities explain the multiplicity of roots when solving trigonometric equations over extended intervals.

对正切，tan(θ + π) = tan θ，所以解每隔 π 弧度重复出现。这些恒等式解释了在扩展区间上解三角方程时出现多个根的原因。

sin(π – θ) = sin θ, cos(-θ) = cos θ, tan(θ + π) = tan θ

Additionally, the relationship sin(θ + 90°) = cos θ means that the cosine graph is exactly the sine graph shifted left by 90°. This is why every cosine graph can be expressed as a sine graph with a phase shift and vice versa.

此外，sin(θ + 90°) = cos θ 表示余弦图像就是正弦图像向左平移 90° 的结果。这就是为什么任何余弦图像都可以写成带相位平移的正弦形式，反之亦然。

9. Inverse Trigonometric Functions | 反三角函数图像

Because sine and cosine are not one-to-one over their entire domains, their inverse functions are defined on restricted intervals. The inverse sine function y = arcsin x has domain -1 ≤ x ≤ 1 and range -π/2 ≤ y ≤ π/2. Its graph is the reflection of the sine curve (restricted) in the line y = x.

由于正弦和余弦在整个定义域上不是一一对应的，其反函数定义在受限区间上。反正弦函数 y = arcsin x 的定义域为 -1 ≤ x ≤ 1，值域为 -π/2 ≤ y ≤ π/2。其图像是（受限的）正弦曲线关于直线 y = x 的反射。

Similarly, y = arccos x has domain -1 ≤ x ≤ 1 and range 0 ≤ y ≤ π. The function y = arctan x has domain all real numbers and range -π/2 < y < π/2, with horizontal asymptotes at y = ±π/2.

类似地，y = arccos x 的定义域为 -1 ≤ x ≤ 1，值域为 0 ≤ y ≤ π。函数 y = arctan x 的定义域为全体实数，值域为 -π/2 < y < π/2，并具有水平渐近线 y = ±π/2。

When sketching inverse trig graphs, remember they are reflections of the original restricted functions. The horizontal asymptotes of arctan are approached but never reached, just like the vertical asymptotes of tan.

画反三角函数图像时，记住它们是原受限函数的反射。arctan 的水平渐近线可以被无限接近但永远不会到达，这与 tan 的竖直渐近线类似。

10. Secant, Cosecant and Cotangent Graphs | 正割、余割和余切图像

Although not primary graphs, the reciprocal functions appear in A-Level syllabus. Since sec x = 1/cos x, the secant graph has vertical asymptotes where cos x = 0, i.e., at x = π/2 + kπ. Between consecutive asymptotes, the curve forms U-shaped branches passing through points where cos x = ±1.

虽然正割、余割和余切不是首要图像，但它们出现在A-Level大纲中。由于 sec x = 1/cos x，正割图像在 cos x = 0 处有竖直渐近线，即 x = π/2 + kπ。相邻渐近线之间形成 U 形分支，经过 cos x = ±1 的点。

Similarly, cosec x = 1/sin x has asymptotes at x = kπ, and cot x = 1/tan x has asymptotes at x = kπ, with a period of π. The graphs of these reciprocals are best sketched by first drawing the original sine, cosine or tangent graph as a dashed reference.

类似地，cosec x = 1/sin x 的渐近线在 x = kπ 处，cot x = 1/tan x 的渐近线也在 x = kπ 处，周期为 π。画这些倒数函数图像的最佳方法是先画出原正弦、余弦或正切图像作为虚线参考。

sec x: asymptotes at x = π/2 + kπ, range (-∞,-1] ∪ [1,∞) | 渐近线 x = π/2 + kπ，值域 (-∞,-1] ∪ [1,∞)
cosec x: asymptotes at x = kπ, range (-∞,-1] ∪ [1,∞) | 渐近线 x = kπ，值域 (-∞,-1] ∪ [1,∞)
cot x: asymptotes at x = kπ, range all real numbers | 渐近线 x = kπ，值域全体实数

11. Applications in Real-World Modelling | 实际建模中的应用

Sine and cosine graphs model periodic phenomena such as tides, sound waves, alternating current and seasonal temperature variation. For example, the height of sea water at a harbour can be represented by H(t) = 4 + 2 sin(πt/6), where t is hours after midnight.

正弦和余弦图像可用于模拟周期性现象，如潮汐、声波、交流电和季节性温度变化。例如，港口海面高度可表示为 H(t) = 4 + 2 sin(πt/6)，其中 t 为午夜后的小时数。

In such models, the amplitude represents the maximum deviation from the mean value, the vertical shift d represents the mean level, and the period represents the time for one complete cycle. Tangent graphs appear in problems involving gradients of rotating objects, such as the angle of a ladder against a rotating wheel.

在此类模型中，振幅表示偏离平均值的最大程度，竖直平移 d 表示平均水平，周期表示完成一次完整循环所需的时间。正切图像出现在涉及旋转物体斜率的题目中，例如梯子与旋转轮之间的角度问题。

When interpreting graphs from real data, always identify the central line first, then measure the amplitude and period. These parameters can then be substituted into y = a sin(bx) + d to create a mathematical model for prediction.

在解读真实数据图像时，先确定中心线，再测量振幅和周期。将这些参数代入 y = a sin(bx) + d 即可建立预测用的数学模型。

12. Exam Tips and Common Mistakes | 考试技巧与常见错误

Common mistakes include using degrees instead of radians in calculus contexts, forgetting the factor b when calculating phase shift, and drawing tangent curves with horizontal asymptotes instead of vertical ones. Always write down the period before sketching.

常见错误包括：在微积分上下文中使用角度制代替弧度制、忘记在计算相位平移时提取因子 b、以及把正切曲线的渐近线画成水平而非竖直。画图前务必先写出周期。

Another frequent error is misidentifying the amplitude of y = -2 cos x, which is 2, not -2. The negative sign only indicates a reflection. For equations that ask for multiple solutions, always add or subtract the period repeatedly within the given domain, and use a sketch to verify each solution.

另一个常见错误是误判 y = -2 cos x 的振幅，它应是 2 而不是 -2。负号只表示反射。当题目要求多个解时，务必在给定定义域内反复加减周期，并利用草图验证每个解。

Function \| 函数	Period \| 周期	Asymptotes \| 渐近线
sin x, cos x	2π	None \| 无
tan x	π	x = π/2 + kπ
sec x, cosec x	2π	sec: x = π/2 + kπ; cosec: x = kπ
cot x	π	x = kπ

Finally, practise sketching all six trig graphs from memory within 30 seconds each. Examiners often award method marks for accurate shapes, asymptotes and intercepts even when transformations are applied. Familiarity with these graphs will also make calculus topics such as differentiating sin x and integrating cos x far more intuitive.

最后，练习在30秒内凭记忆画出全部六个三角函数的图像。考官通常会根据形状、渐近线和交点的准确性给方法分，即使题目包含了变换。熟练掌握这些图像还能让微积分内容（如对 sin x 求导、对 cos x 积分）变得更加直观。

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A-Level Maths: Graphs of Sine, Cosine and Tangent | A-Level数学：正弦、余弦和正切图像