📚 Graphs of sine, cosine and tangent | 正弦、余弦和正切函数的图像
a ranging from sound waves to the motion of a pendulum, and they form the foundation for more advanced topics such as trigonometric equations, identities, and calculus of trigonometric functions. In this article, we will explore the shapes, key properties, and transformations of y = sin x, y = cos x and y = tan x, with all angles measured in radians.
理解三个基本三角函数的图像是A-Level数学的核心技能。这些曲线可以模拟从声波到单摆运动的周期现象，并为三角方程、三角恒等式和三角函数微积分等更高阶的主题奠定基础。本文将探索 y = sin x、y = cos x 和 y = tan x 的形状、关键性质及其图像变换，所有角度均以弧度度量。
1. Introduction to Trigonometric Graphs | 三角函数图像简介
e defined for real numbers x, typically representing an angle in radians. Unlike the right‑ strict angles to acute values, the graphs extend infinitely in both directions along the x‑ efined for all real x, while the tangent function has domain restrictions where cos x = 0.
三角函数 sin x、cos x 和 tan x 是为实数 x 定义的，通常表示以弧度为单位的角度。与仅适用于锐角的直角三角形定义不同，这些图像沿 x 轴向两个方向无限延伸，展现出它们的周期性。正弦和余弦函数对所有实数 x 都有定义，而正切函数在 cos x = 0 处存在定义域限制。
The coordinate plane for trigonometric graphs uses x as the angle (in radians) and y as the function value. This means that the familiar properties—period, amplitude, intercepts, and asymptotes—can be read directly from the graph. A solid grasp of the basic curves is essential before tackling transformations such as stretches, reflections, and translations.
三角函数图像所在的坐标平面以 x 为角度（弧度），y 为函数值。这意味着图像的周期、振幅、截距和渐近线等常见性质可以直接从图上读出来。在解决伸缩、反射和平移等图像变换问题之前，牢固掌握基础曲线是至关重要的。
2. The Graph of y = sin x | y = sin x 的图像
/2, returns to 0 at x = π, reaches a minimum of −1 at x = 3π/2, and completes one full cycle at x = 2π. Because sine is an odd function, the graph is symmetric about the origin—rotating it by 180° about the origin leaves the graph unchanged.
y = sin x 的图像是一条光滑、连续的波形，在 −1 和 1 之间振荡。曲线从原点 (0,0) 出发，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，并在 x = 2π 处完成一个完整周期。由于正弦是奇函数，图像关于原点对称——绕原点旋转 180° 后图像不变。
This wave pattern repeats every 2π units along the x‑ e period of the sine function. The portion of the graph for 0 ≤ x ≤ 2π is often called the 'fundamental cycle' rotational symmetry of order 2 about the origin and also about points (π,0), (−π,0), etc.
这个波形沿 x 轴每 2π 个单位重复一次，这就是正弦函数的周期。通常将 0 ≤ x ≤ 2π 范围内的图像称为“基本周期”。正弦曲线关于原点具有 2 阶旋转对称性，同时也关于 (π,0)、(−π,0) 等点具有旋转对称性。
3. Properties of the Sine Curve | 正弦曲线的性质
n be summarised neatly in a table. It is important to express these properties using both degrees (for conceptual comfort) and radians, as radian measure is the default in A-Level calculus and graph work.
y = sin x 的关键特性可以用一张表格清晰地总结。同时使用度数和弧度表达这些性质很重要——度数便于理解概念，但弧度是 A-Level 微积分和图像学习中的默认度量。
| Property | Value / Description |
|---|---|
| Domain | All real numbers (ℝ) |
| Range | [−1, 1] |
| Period | 2π (360°) |
| Amplitude | 1 |
| Maximum points | (π/2 + 2nπ, 1) for n ∈ ℤ |
| Minimum points | (−π/2 + 2nπ, −1) or (3π/2 + 2nπ, −1) |
| x‑intercepts (roots) | (nπ, 0) for n ∈ ℤ |
| y‑intercept | (0, 0) |
| Symmetry | Odd function; rotational symmetry about origin |
= sin x this is 1. When the function is transformed to y = a sin x, the amplitude becomes |a| positions at multiples of π are extremely useful when solving equations such as sin x = 0.
周期函数的振幅是其最大值与最小值之间距离的一半；对于 y = sin x，振幅为 1。当函数变为 y = a sin x 时，振幅为 |a|。位于 π 的整数倍处的零点在求解诸如 sin x = 0 的方程时非常有用。
4. The Graph of y = cos x | y = cos x 的图像
ximum point (0,1). It then descends through (π/2, 0), reaches a minimum at (π, −1), rises back through (3π/2, 0), and returns to a maximum at (2π, 1).
y = cos x 的图像同样是一条在 −1 和 1 之间变化的光滑波形，但起始位置不同。在 x = 0 处，cos 0 = 1，因此图像从最大值点 (0,1) 开始。接着它经过 (π/2, 0) 下降，在 (π, −1) 处达到最小值，再经过 (3π/2, 0) 上升，并在 (2π, 1) 处回到最大值。
= sin(x + π/2). Therefore, these two curves are identical in shape but displaced along the x‑axis by π/2.
与正弦类似，余弦函数的周期为 2π，因此图像每 2π 重复一次。余弦图像实际上是正弦图像的水平平移：cos x = sin(x + π/2)。所以这两条曲线形状相同，只是沿 x 轴平移了 π/2。
5. Properties of the Cosine Curve | 余弦曲线的性质
The cosine function shares many features with sine, but its even symmetry and different intercept positions are important distinctions. The table below provides a concise reference.
余弦函数与正弦函数有许多共同特征，但其偶函数对称性和不同的截距位置是需要区分的重要特性。下表提供了简明参考。
| Property | Value / Description |
|---|---|
| Domain | ℝ |
| Range | [−1, 1] |
| Period | 2π |
| Amplitude | 1 |
| Maximum points | (2nπ, 1) for n ∈ ℤ |
| Minimum points | ((2n+1)π, −1) for n ∈ ℤ |
| x‑intercepts (roots) | (π/2 + nπ, 0) for n ∈ ℤ |
| y‑intercept | (0, 1) |
| Symmetry | Even function; reflection symmetry in y‑axis |
Being an even function means that cos(−x) = cos x, which is evident from the mirror symmetry of the graph across the y‑ ntly exploited when solving equations such as cos θ = cos α, giving solutions θ = 2nπ ± α.
余弦是偶函数，这意味着 cos(−x) = cos x，从图像关于 y 轴的镜像对称可以明显看出。这一性质在求解诸如 cos θ = cos α 的方程时经常被利用，其解为 θ = 2nπ ± α。
6. The Graph of y = tan x | y = tan x 的图像
n is undefined, resulting in vertical asymptotes. These occur at x = π/2 + nπ for all integers n.
正切函数的行为与正弦、余弦大不相同，因为它被定义为 tan x = sin x / cos x。当 cos x = 0 时，正切函数无定义，因此产生垂直渐近线。这些渐近线出现在 x = π/2 + nπ（n 为所有整数）。
y = tan x increases from −∞ to +∞, passing through the origin at (0,0) and crossing the x‑axis at every integer multiple of π. The shape is not a wave but a sequence of separate, S‑shaped branches. The graph is symmetric about the origin, making tan x an odd function.
在相邻渐近线之间，y = tan x 的图像从 −∞ 增加到 +∞，经过原点 (0,0)，并在 π 的每个整数倍处与 x 轴相交。它的形状不是连续的波形，而是一系列分离的 S 形分支。图像关于原点对称，因此 tan x 是奇函数。
7. Properties of the Tangent Curve | 正切曲线的性质
Because tan x has no maximum or minimum values, it does not possess an amplitude. The key features are its period, asymptotes, and intercepts, summarised below.
由于 tan x 没有最大值或最小值，因此它不具备振幅。其关键特征在于周期、渐近线和截距，总结如下。
| Property | Value / Description |
|---|---|
| Domain | x ∈ ℝ, x ≠ π/2 + nπ |
| Range | ℝ (all real numbers) |
| Period | π (180°) |
| Asymptotes | x = π/2 + nπ, n ∈ ℤ |
| x‑intercepts (roots) | (nπ, 0) for n ∈ ℤ |
| y‑intercept | (0, 0) |
| Symmetry | Odd function; rotational symmetry about origin |
for all x in its domain. The pattern of asymptotes every π units means that when solving equations involving tan, you often add multiples of π to a principal solution.
注意 tan x 的周期是 π，而非 2π。这是因为对于定义域内的所有 x，tan(x + π) = tan x。渐近线每隔 π 个单位出现一次的模式意味着，在解含有正切的方程时，通常需要在主解的基础上加上 π 的整数倍。
8. Transformations of Sine and Cosine Graphs | 正弦与余弦图像的变换
h parameter controls a specific graphical transformation. It is essential to interpret these correctly, especially the horizontal stretch and phase shift.
在 A-Level 考试中，你经常会见到形如 y = a sin(bx + c) + d 或 y = a cos(bx + c) + d 的函数。每个参数控制着一种特定的图像变换。正确理解这些变换非常重要，尤其是水平伸缩和相位平移。
- |a| – vertical stretch / amplitude: The graph's amplitude becomes |a|. If a is negative, there is also a reflection in the x‑axis.
- |b| – horizontal stretch / period change: The period becomes 2π/|b|. For b > 1 the graph squashes horizontally; for 0 < b < 1 it stretches.
- c – bx + c) = sin[b(x + c/b)], the shift is −c/b (to the left if positive inside the bracket).
- d – vertical translation: The whole graph moves up (d > 0) or down (d < 0) by d units.
- |a| —— 垂直伸缩 / 振幅：图像的振幅变为 |a|。若 a 为负，还会附加关于 x 轴的反射。
- |b| —— 水平伸缩 / 周期变化：周期变为 2π/|b|。b > 1 时图像水平压缩；0 < b < 1 时水平拉伸。
- c —— 水平平移（相位平移）：写成 sin(bx + c) = sin[b(x + c/b)]，平移量为 −c/b（括号内为加号时向左平移）。
- d —— 垂直平移：整个图像向上 (d > 0) 或向下 (d < 0) 移动 d 个单位。
, and is shifted up by 1. The minimum value will be −3 + 1 = −2 and the maximum 3 + 1 = 4.
例如，函数 y = 3 sin(2x − π/3) + 1 的振幅为 3，周期为 2π/2 = π，相位向右平移 (π/3)/2 = π/6，并向上平移 1 个单位。其最小值将是 −3 + 1 = −2，最大值为 3 + 1 = 4。
9. Transformations of the Tangent Graph | 正切图像的变换
cause there is no amplitude, the parameter a instead causes a vertical stretch about the x‑axis. The function y = a tan(bx + c) + d has asymptotes and period determined by b and c.
y = tan x 的变换遵循类似的原理，但由于正切没有振幅，参数 a 会引起关于 x 轴的垂直拉伸。函数 y = a tan(bx + c) + d 的渐近线和周期则由 b 和 c 决定。
The period of y = tan(bx + c) is π/|b| c/b, and the vertical translation is d. Note that a negative a reflects the graph in the x‑axis, which can be useful for sketching.
y = tan(bx + c) 的周期是 π/|b|。其垂直渐近线可通过求解 bx + c = π/2 + nπ 得到。水平平移量为 −c/b，垂直平移量为 d。注意，a 为负时会关于 x 轴反射图像，这在草图绘制时很有用。
For instance, y = 2 tan(x/2) has period π/(1/2) = 2π, and asymptotes at x/2 = π/2 + nπ ⇒ retched vertically by a factor of 2 and stretched horizontally, making the branches wider.
例如，y = 2 tan(x/2) 的周期为 π/(1/2) = 2π，渐近线在 x/2 = π/2 + nπ ⇒ x = π + 2nπ。图像被垂直拉伸到原来的 2 倍，并水平拉宽，使得每个分支更加展开。
10. Using Graphs to Solve Trigonometric Equations | 利用图像解三角方程
t branches and a horizontal line y = k, you can read off all solutions within a given interval.
三角函数图像为求解诸如 sin x = k、cos x = k 或 tan x = k 的方程提供了强大的可视化方法。通过画出基本波形或正切分支以及水平线 y = k，你可以读出在给定区间内的所有解。
ine curve, sin(π − x) = sin x, you can justify the second solution. For tangent, once you find one solution, you add multiples of the period, π, to generate all others.
例如，求解 sin x = 0.5 在 0 ≤ x ≤ 2π 内的解，可以画出 y = sin x 和直线 y = 0.5。交点位于 x = π/6 和 x = 5π/6。利用正弦曲线的对称性 sin(π − x) = sin x，你可以验证第二个解。对于正切，一旦找到一个解，你只需加上周期的整数倍 π，即可生成所有其他解。
e left, and tan x = sin x/cos x explains why the tangent graph has zeros where sine does and asymptotes where cosine is zero.
这种图像方法还能加深对相关恒等式的理解。平移恒等式 cos x = sin(x + π/2) 可以通过将正弦曲线向左平移看出来，而 tan x = sin x/cos x 则解释了为什么正切图像在正弦为零处有零点，在余弦为零处有渐近线。
11. Symmetry and Identities from the Graphs | 从图像看对称性与恒等式
dentities become visually obvious when studied on the graphs. For instance, the sine curve's rotational symmetry about the origin gives sin(−x) = −sin x. The cosine graph's mirror symmetry in the y‑axis confirms cos(−x) = cos x.
许多三角恒等式在图像上能一目了然。例如，正弦曲线关于原点的旋转对称性给出 sin(−x) = −sin x。余弦图像关于 y 轴的镜像对称则证实了 cos(−x) = cos x。
dentities such as sin(x + 2π) = sin x and tan(x + π) = tan x, which are essential when finding general solutions.
此外，零点和峰值揭示了补角关系：sin(π − x) = sin x 和 cos(2π − x) = cos x。周期性则支撑了诸如 sin(x + 2π) = sin x 和 tan(x + π) = tan x 等恒等式，这些在求解通解时至关重要。
Being able to 'read' d π/6, so sin(7π/6) = −sin(π/6) = −1/2. This combines the idea of periodicity and the sine's symmetry about the origin.
能够从草图中“读出”这些对称性可以在考试中节省时间。当要求计算 sin(7π/6) 时，你可以参考图像：7π/6 比 π/6 多出 π，因此 sin(7π/6) = −sin(π/6) = −1/2。这结合了周期性和正弦关于原点的对称性。
12. Summary and Exam Tips | 总结与考试技巧
Mastering the graphs of sine, cosine and tangent means knowing their shapes, periods, amplitudes (where applicable), symmetry, and key points. Always work in radians unless the question specifies degrees, and label axes carefully when sketching. For transformed functions, identify the sequence of transformations by rewriting the function in the form a f(b(x + h)) + k.
掌握正弦、余弦和正切的图像意味着要熟悉它们的形状、周期、振幅（如果适用）、对称性以及关键点。除非题目特别说明使用度数，否则一律使用弧度，并在画图时仔细标注坐标轴。对于经过变换的函数，通过将函数重写为 a f(b(x + h)) + k 的形式来识别变换的顺序。
rdering the horizontal transformations. Remember: the period of sin(bx) and cos(bx) is 2π/|b|, while that of tan(bx) is π/|b|. Asymptotes for tan occur where cos(bx + c) = 0, and the phase shift is always −c/b.
常见的考试错误包括混淆正切的周期与正弦、余弦的周期，错误地放置渐近线，以及弄错水平变换的次序。请记住：sin(bx) 和 cos(bx) 的周期是 2π/|b|，而 tan(bx) 的周期是 π/|b|。正切的渐近线出现在 cos(bx + c) = 0 处，且相位平移永远是 −c/b。
Finally, practice sketching graphs quickly but accurately. Use the basic cycle and key points as a framework, then apply stretches and shifts. This will help you solve equations, find domains and ranges, and tackle calculus applications such as modelling with trigonometric functions.
最后，要练习快速而准确地绘制图像。以基本周期和关键点为框架，然后施加拉伸和平移。这将帮助你求解方程、确定定义域和值域，并应对诸如三角函数建模等微积分应用。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图