📚 Graphs of Sine, Cosine and Tangent | 正弦、余弦和正切图像
The graphs of the trigonometric functions sine, cosine and tangent are fundamental in IGCSE and A Level Mathematics. They model periodic behaviour - waves, rotations, tides and alternating currents - and they appear constantly in problem-solving questions. Understanding their shapes, periods, amplitudes, asymptotes and symmetries allows you to sketch them accurately, solve trigonometric equations and apply transformations with confidence.
三角函数正弦、余弦和正切的图像是 IGCSE 与 A Level 数学的基础。它们用于模拟周期行为——波、旋转、潮汐与交流电——并频繁出现在解题类题目中。理解它们的形状、周期、振幅、渐近线与对称性，使你能够准确作图、求解三角方程并自信地应用变换。
1. The Unit Circle and Trigonometric Definitions | 单位圆与三角函数定义
The unit circle is the circle of radius 1 centred at the origin of the coordinate plane. If an angle θ is measured anticlockwise from the positive x-axis, the point where the terminal arm of the angle meets the circle has coordinates (cos θ, sin θ). Every value of sine and cosine therefore comes directly from a coordinate on this circle.
单位圆是坐标平面上以原点为圆心、半径为 1 的圆。若角 θ 从正 x 轴逆时针测量，则角的终边与该圆的交点坐标为 (cos θ, sin θ)。因此每一个正弦值与余弦值都直接来自这个圆上一点的坐标。
Tangent is defined as the ratio of these two coordinates, which links all three functions together:
正切定义为这两个坐标的比值，它把三个函数联系在一起：
tan θ = sin θ ÷ cos θ
e repeat every 360° as well, while tangent repeats every 180° since tan(θ + 180°) = tan θ.
该比值解释了每个函数的定义域与值域。由于圆每 360° 重复一次，正弦与余弦也每 360° 重复一次；而正切每 180° 重复，因为 tan(θ + 180°) = tan θ。
The unit circle also shows the minimum and maximum values. sin θ and cos θ always lie between -1 and 1 inclusive, while tan θ can take any real value, because the ratio becomes infinitely large as cos θ approaches zero. This is exactly why the tangent graph has vertical asymptotes.
单位圆还显示了最小值与最大值。sin θ 与 cos θ 总是介于 -1 与 1 之间（含端点），而 tan θ 可以取任何实数值，因为当 cos θ 趋近于零时该比值可以无限增大。这正是正切图像出现竖直渐近线的原因。
- Range of sine and cosine: -1 ≤ sin θ ≤ 1 and -1 ≤ cos θ ≤ 1 | 正弦与余弦的值域：-1 ≤ sin θ ≤ 1，-1 ≤ cos θ ≤ 1
- Range of tangent: tan θ ∈ ℝ | 正切的值域：tan θ ∈ ℝ（一切实数）
- Periods: sin and cos 360°, tan 180° | 周期：sin 与 cos 为 360°，tan 为 180°
2. The Graph of y = sin x | y = sin x 的图像
The sine function produces a smooth, continuous wave. It passes through the origin (0, 0), rises to a maximum of 1 at 90°, returns to 0 at 180°, drops to a minimum of -1 at 270°, and returns to the origin's height, 0, at 360°. The shape then repeats identically every 360°.
正弦函数产生一条光滑、连续的波形。它穿过原点 (0, 0)，在 90° 时升至最大值 1，在 180° 时回到 0，在 270° 时降至最小值 -1，并在 360° 时回到 0。此后该形状每 360° 完全相同地重复。
The key coordinates of one complete cycle are summarised below:
一个完整周期的关键坐标总结如下：
| x | 0° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|
| sin x | 0 | 1 | 0 | -1 | 0 |
s at 90° and 270°. The curve has rotational symmetry of order 2 about the origin, which is the graph's way of showing sin(-x) = -sin x.
正弦曲线常被描述为"从中间出发向上走"。注意在 0°、180° 与 360° 处有三个 x 轴交点，在 90° 与 270° 处有两个转折点。该曲线关于原点具有二阶旋转对称性，这正是图像表达 sin(-x) = -sin x 的方式。
3. The Graph of y = cos x | y = cos x 的图像
0, falls to 0 at 90°, reaches -1 at 180°, returns to 0 at 270°, and returns to 1 at 360°.
余弦曲线与正弦曲线形状完全相同，只是向左平移了 90°。它在 x = 0 时从最大值 1 出发，在 90° 降到 0，在 180° 达到 -1，在 270° 回到 0，并在 360° 回到 1。
| x | 0° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|
| cos x | 1 | 0 | -1 | 0 | 1 |
s x. This symmetry is one of the most useful tools in trigonometry, and it explains why the curve looks identical when you read it from right to left.
由于余弦图像关于 y 轴对称（自身镜像重合），余弦被称为偶函数：cos(-x) = cos x。这一对称性是三角学中最有用的工具之一，也解释了为什么从右向左读该曲线时它看起来完全一样。
(x + 90°). The cosine curve is simply the sine curve moved 90° to the left along the x-axis.
直接比较这两条曲线可得到重要恒等式 cos x = sin(x + 90°)。余弦曲线不过是正弦曲线沿 x 轴向左平移 90° 的结果。
4. The Graph of y = tan x | y = tan x 的图像
rom -∞ to +∞ as x increases through an interval of 180°, and the graph repeats every 180°.
正切图像的性质与正弦、余弦完全不同。它不是波形，而是一系列彼此分离的分支。每一分支在 x 增加 180° 的区间内从 -∞ 上升到 +∞，整个图像每 180° 重复一次。
t is at 90°, 270°, 450° and so on. At these values the graph has vertical asymptotes, and the curve approaches them more and more steeply without ever touching them.
因为 tan x = sin x ÷ cos x，所以当 cos x = 0 时函数无定义，即 90°、270°、450° 等处。在这些值处图像有竖直渐近线，曲线越来越陡地趋近它们，却永不相交。
| x | 0° | 45° | 90° | 135° | 180° |
|---|---|---|---|---|---|
| tan x | 0 | 1 | undefined | -1 | 0 |
s no amplitude, and is an odd function, so tan(-x) = -tan x. The general equation of its asymptotes is x = 90° + 180°n, where n is any integer.
需要记住的关键特征：正切图像过原点，在每个 180° 的整数倍处与 x 轴相交，没有最大值也没有最小值，没有振幅，并且是奇函数，因此 tan(-x) = -tan x。其渐近线的一般方程为 x = 90° + 180°n，其中 n 为任意整数。
5. Amplitude, Period and the Midline | 振幅、周期与中线
Three vocabulary terms describe every periodic graph precisely. The amplitude is the distance from the midline to a maximum (or minimum) value. The period is the length of the x-axis interval containing one complete cycle. The midline is the horizontal line halfway between the highest and lowest points.
有三个术语可精确描述任何周期图像。振幅是从中线到最大值（或最小值）的距离。周期是包含一个完整周期的 x 轴区间长度。中线是位于最高点与最低点正中间的水平直线。
tude at all, because the branches are unbounded; its period is 180° and it has no midline in the usual sense.
对于 y = sin x 与 y = cos x，振幅为 1，周期为 360°，中线为 y = 0。对于 y = tan x 则根本没有振幅，因为各分支无界；其周期为 180°，并且通常意义下没有中线。
| Function 函数 | Period 周期 | Amplitude 振幅 | Range 值域 |
|---|---|---|---|
| y = sin x | 360° | 1 | -1 ≤ y ≤ 1 |
| y = cos x | 360° | 1 | -1 ≤ y ≤ 1 |
| y = tan x | 180° | none 无 | all real numbers 一切实数 |
A useful check in an exam is to count the number of complete waves drawn between two values of x, then divide the interval length by that number to confirm the period.
考试中的一个实用检验方法是：数出两个 x 值之间画了多少个完整波形，再用区间长度除以该数量来验证周期。
6. Transformations: y = a sin(bx) + c | 变换：y = a sin(bx) + c
our pieces of information. The factor a stretches the graph vertically and gives amplitude |a| 360° ÷ b. The constant c shifts the whole graph vertically and defines the midline y = c.
一般正弦曲线 y = a sin(bx) + c 包含四类信息。系数 a 使图像竖直拉伸，给出振幅 |a|。系数 b 使图像水平压缩，给出周期 360° ÷ b。常数 c 使整条曲线竖直平移，并确定中线 y = c。
The maximum and minimum values follow immediately from the amplitude and midline:
最大值与最小值可直接由振幅与中线得到：
maximum = c + |a| minimum = c - |a|
bx is moved α to the right. The cosine graph is the special case α = 90° relative to sine.
当括号内含有常数时，出现水平平移（相位移动），例如 y = a sin b(x - α)。此时 y = a sin bx 的图像向右平移 α。余弦图像正是正弦图像中 α = 90° 的特殊情形。
- y = 3 sin 2x + 1: amplitude 3, period 180°, midline y = 1, maximum 4, minimum -2 | 振幅 3，周期 180°，中线 y = 1，最大值 4，最小值 -2
- y = -2 cos x: amplitude 2, period 360°, midline y = 0, graph reflected in the x-axis | 振幅 2，周期 360°，中线 y = 0，图像关于 x 轴翻转
- y = tan 2x: period 90°, asymptotes at x = 45° + 90°n | 周期 90°，渐近线为 x = 45° + 90°n
When a is negative the wave is turned upside down, but the amplitude is still written as a positive number. This is a very common source of lost marks.
当 a 为负数时波形上下颠倒，但振幅仍写作正数。这是最常见的失分点之一。
7. Symmetries and Identities from the Graphs | 图像中的对称性与恒等式
The cosine graph has reflection symmetry in the y-axis, which gives cos(-x) = cos x. The tangent graph also has rotational symmetry about the origin, so tan(-x) = -tan x.
每个图像的对称性都对应一条三角恒等式。正弦图像关于原点有旋转对称性，给出 sin(-x) = -sin x。余弦图像关于 y 轴有反射对称性，给出 cos(-x) = cos x。正切图像同样关于原点有旋转对称性，故 tan(-x) = -tan x。
ertical line x = 90° gives sin(180° - x) = sin x, and comparing the two waves gives cos x = sin(90° - x).
将图像结合起来阅读还可得到互余与互补关系。例如正弦曲线关于竖直线 x = 90° 的对称性给出 sin(180° - x) = sin x，而比较两条波可得到 cos x = sin(90° - x)。
- sin(180° - x) = sin x, but cos(180° - x) = -cos x | sin(180° - x) = sin x，但 cos(180° - x) = -cos x
- sin(180° + x) = -sin x and cos(180° + x) = -cos x | sin(180° + x) = -sin x，cos(180° + x) = -cos x
- sin(360° - x) = -sin x is the reflection property of the wave | sin(360° - x) = -sin x 是波形的反射性质
These symmetries let you find every solution of an equation in a given interval once you have found one solution, without drawing the whole graph again.
这些对称性使你一旦找到一个解，就能在给定区间内找出方程的全部解，而无需重新画出完整图像。
8. Solving Trigonometric Equations Graphically | 用图像解三角方程
l line y = 0.5. The intersection points give the solutions directly: x = 30° and x = 150°.
要求解如 sin x = 0.5（0° ≤ x ≤ 360°）这样的方程，可先画出曲线 y = sin x，再画水平线 y = 0.5。交点直接给出解：x = 30° 与 x = 150°。
The general solution pattern follows from the wave' - α + 360°n. If cos θ = cos α, then θ = ±α + 360°n. If tan θ = tan α, then θ = α + 180°n.
一般解的规律来自波形的对称性。若 sin θ = sin α，则 θ = α + 360°n 或 θ = 180° - α + 360°n。若 cos θ = cos α，则 θ = ±α + 360°n。若 tan θ = tan α，则 θ = α + 180°n。
- sin x = 0.5 gives x = 30°, 150° in one full cycle | sin x = 0.5 在一个完整周期内给出 x = 30°、150°
- cos x = 0.5 gives x = 60°, 300° in one full cycle | cos x = 0.5 在一个完整周期内给出 x = 60°、300°
- tan x = 1 gives x = 45°, 225° in one full cycle | tan x = 1 在一个完整周期内给出 x = 45°、225°
Always check the required interval. A question asking for solutions between -180° and 540° expects four or more answers, and the graph will show exactly how many intersections exist.
务必检查题目要求的区间。若题目要求 -180° 到 540° 之间的解，通常需要四个或更多答案，而图像会准确显示有多少个交点。
9. Radians and Graph Interpretation | 弧度制与图像解读
ne curves have period 2π, and the tangent curve has period π. The shape of every graph is unchanged; only the scale of the horizontal axis is different.
在 A Level 中，同样的图像以弧度为 x 轴单位绘制。因为 π 弧度 = 180°，正弦与余弦曲线的周期为 2π，正切曲线的周期为 π。所有图像的形状
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