📚 Graphs of sine, cosine and tangent | 正弦、余弦与正切函数图像
In Edexcel A-Level Mathematics, the graphs of y = sin x, y = cos x and y = tan x are studied with x measured in radians unless a question states degrees. They are periodic functions, so their graphs repeat at regular intervals and can be used to solve equations and model oscillations.
在 Edexcel A-Level 数学中，y = sin x、y = cos x 和 y = tan x 的图像通常以弧度制表示 x，除非题目明确使用角度制。它们都是周期函数，图像按固定间隔重复，因此可用来解方程和描述振动现象。
The sine and cosine functions are defined for all real x, while the tangent function is undefined wherever cos x = 0.
正弦函数和余弦函数的定义域为全体实数，而正切函数在 cos x = 0 处无定义。
1. The core trigonometric ratios and their graph domains | 核心三角比及其图像定义域
The three trigonometric functions are built from the ratios of sides in a right-angled triangle, but their graphs extend these ratios to all real angles using the unit circle or periodic extension.
三个三角函数来自直角三角形中边的比值，但它们的图像通过单位圆或周期延拓，将这些比值推广到全体实数角。
For y = sin x and y = cos x, every real x-value has exactly one y-value. For y = tan x, the function is undefined at x = π/2 + nπ, where n is an integer, because division by zero occurs.
对 y = sin x 和 y = cos x 来说，每个实数 x 都有唯一的 y 值。对 y = tan x 来说，函数在 x = π/2 + nπ（n 为整数）处无定义，因为此时出现了除以零的情况。
You must be able to sketch all three graphs for at least one full period and to label key points exactly.
你必须能够至少画出一个完整周期内这三个函数的图像，并准确标出关键点。
2. Graph of y = sin x: wave from the origin | y = sin x 的图像：从原点出发的波
The graph of y = sin x starts at the origin with value 0, rises to a maximum of 1 at x = π/2, returns to 0 at x = π, falls to a minimum of −1 at x = 3π/2, and returns to 0 at x = 2π.
y = sin x 的图像从原点出发，函数值为 0；在 x = π/2 处升至最大值 1；在 x = π 处回到 0；在 x = 3π/2 处降至最小值 −1；在 x = 2π 处再次回到 0。
Its fundamental period is 2π, so sin(x + 2π) = sin x for all x. The graph is rotationally symmetric about the origin, making sine an odd function: sin(−x) = −sin x.
它的基本周期为 2π，因此对任意 x 都有 sin(x + 2π) = sin x。图像关于原点中心对称，所以正弦是奇函数：sin(−x) = −sin x。
The zeros of sin x occur at x = nπ, where n is any integer. These are the points where the graph crosses the x-axis.
sin x 的零点位于 x = nπ（n 为任意整数），这些点是图像与 x 轴的交点。
3. Graph of y = cos x: the same wave shifted | y = cos x 的图像：平移后的同一波形
The graph of y = cos x has the same shape as the sine graph, but it is translated horizontally to the left by π/2. It begins at (0,1), falls to 0 at x = π/2, reaches −1 at x = π, rises to 0 at x = 3π/2, and returns to 1 at x = 2π.
y = cos x 的图像与正弦图像形状相同，只是向左平移了 π/2。它从点 (0,1) 开始，在 x = π/2 处降到 0，在 x = π 处达到 −1，在 x = 3π/2 处升到 0，在 x = 2π 处回到 1。
The cosine graph is symmetric about the y-axis, so cos(−x) = cos x; it is an even function. Its period is also 2π.
余弦图像关于 y 轴对称，因此 cos(−x) = cos x；它是偶函数。周期同样为 2π。
The relationship cos x = sin(x + π/2) shows that the cosine graph is simply the sine graph shifted left by π/2. This identity is useful when transforming graphs.
恒等式 cos x = sin(x + π/2) 表明余弦图像就是正弦图像向左平移 π/2 得到的结果。这一关系在图像变换中非常有用。
4. Graph of y = tan x: period, zeros and asymptotes | y = tan x 的图像：周期、零点与渐近线
The tangent graph repeats every π, not every 2π. It passes through the origin and is increasing on every interval between consecutive vertical asymptotes.
正切图像每 π 重复一次，而不是每 2π。它经过原点，并在相邻两条竖直渐近线之间的每个区间上递增。
Vertical asymptotes occur at x = π/2 + nπ, where n is an integer, because tan x = sin x / cos x and cos x = 0 at these points. The graph approaches these lines but never touches them.
竖直渐近线出现在 x = π/2 + nπ（n 为整数），因为 tan x = sin x / cos x，而在这些点 cos x = 0。图像无限接近这些直线但永不接触。
The zeros of tan x are at x = nπ, exactly where sin x = 0. The graph has no maximum or minimum value, so it has no amplitude.
正切函数的零点位于 x = nπ，也就是 sin x = 0 的位置。图像没有最大值或最小值，因此没有振幅。
5. Amplitude, period, and key features | 振幅、周期与关键特征
For y = a sin(bx + c) + d, the amplitude is |a|, the vertical distance from the centre line to a maximum or minimum. The basic sine and cosine graphs have amplitude 1.
对 y = a sin(bx + c) + d 来说，振幅是 |a|，即中心线到最大值或最小值的竖直距离。基本正弦和余弦图像的振幅都是 1。
The period of y = sin(bx) or y = cos(bx) is 2π/|b|. The tangent graph y = tan(bx) has period π/|b|.
y = sin(bx) 或 y = cos(bx) 的周期为 2π/|b|；正切图像 y = tan(bx) 的周期为 π/|b|。
| Function | Amplitude | Period | Zeros | Symmetry |
|---|---|---|---|---|
| sin x | 1 | 2π | nπ | Odd |
| cos x | 1 | 2π | π/2 + nπ | Even |
| tan x | None | π | nπ | Odd |
Exact values at standard angles are essential for accurate sketches. For example, sin(π/6) = 1/2, cos(π/4) = √2/2 and tan(π/3) = √3.
标准角的精确值对于准确画图至关重要。例如 sin(π/6) = 1/2，cos(π/4) = √2/2，tan(π/3) = √3。
6. Using radians and degrees | 弧度制与角度制的使用
Edexcel exam questions usually use radians for trigonometric graphs and calculus, but they may include degrees in geometric contexts. You must label the x-axis correctly: 0, π/2, π, 3π/2, 2π when working in radians.
Edexcel 考试题中，三角图像和微积分通常使用弧度制，但几何情境下也可能使用角度制。必须正确标注 x 轴：使用弧度时标为 0、π/2、π、3π/2、2π。
To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. Exact values such as sin(π/3) = √3/2 must be known without a calculator.
将角度转换为弧度，乘以 π/180；将弧度转换为角度，乘以 180/π。必须不借助计算器掌握精确值，如 sin(π/3) = √3/2。
In a sketch, the horizontal scale changes completely if you switch from radians to degrees. A period of 2π in radians corresponds to 360° in degrees.
在草图中，如果从弧度制切换为角度制，水平刻度会完全不同。弧度制下的周期 2π 对应角度制下的 360°。
7. Transformations of trigonometric graphs | 三角函数的图像变换
The general transformed sine or cosine graph is y = a sin(bx + c) + d. The value a stretches the graph vertically by a factor of |a| and reflects it in the x-axis if a is negative.
一般变换后的正弦或余弦图像为 y =
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