📚 The Cosine Function | 余弦函数
The cosine function is one of the foundational trigonometric functions in IB Mathematics, central to modelling waves, circular motion, and periodic behaviour. Understanding its graph, properties, and transformations is essential for success in both analysis and applications, as it appears in topics ranging from triangle solving to calculus and real-world modelling.
余弦函数是 IB 数学中基本的三角函数之一，对于波动、圆周运动和周期行为的建模至关重要。理解它的图像、性质和变换是确保在分析与应用领域取得好成绩的关键，因为它出现在从解三角形到微积分再到现实世界建模的各个主题中。
1. Definition via the Unit Circle | 基于单位圆的定义
On the unit circle, the cosine of an angle θ is defined as the x-coordinate of the point where the terminal side of the angle intersects the circle. For any real number θ measured in radians, you can imagine wrapping the angle around the circle; cos θ gives the horizontal displacement of that point from the origin.
在单位圆上，角 θ 的余弦值定义为该角终边与单位圆交点的 x 坐标。对于以弧度度量的任意实数 θ，你可以想象将角度绕在圆上；cos θ 给出了该点相对于原点的水平位移。
This definition extends the right-triangle ratio (adjacent/hypotenuse) to all real numbers, explaining why cos θ can be negative for obtuse angles and is periodic with period 2π.
这一定义将直角三角形的比值（邻边/斜边）推广到了所有实数，解释了为何 cos θ 对钝角可以为负，并且周期为 2π。
2. Graph of y = cos x | 余弦函数 y = cos x 的图像
minimum of -1 at x = π, returns to zero at x = 3π/2, and completes a full cycle at x = 2π.
y = cos x 的图像是一条光滑、连续的波浪线，在 -1 和 1 之间振荡。当 x = 0 时它从最大值 1 开始，在 x = π/2 处与 x 轴相交，在 x = π 处达到最小值 -1，于 x = 3π/2 处再次过零，并在 x = 2π 处完成一个完整周期。
Because the waveform is symmetric about the y-axis, the graph repeats itself every 2π units along the x-axis. The key points (0,1), (π/2,0), (π,-1), (3π/2,0), and (2π,1) form one cycle and are often used to sketch transformations.
由于波形关于 y 轴对称，图像沿 x 轴每隔 2π 单位重复一次。关键点 (0,1)、(π/2,0)、(π,-1)、(3π/2,0) 和 (2π,1) 构成一个周期，常用于绘制变换后的图像。
3. Amplitude, Period, and Phase Shift | 振幅、周期与相位移
y = A cos(B(x - C)) + D, where the parameters control the shape and position of the graph. |A| is the amplitude (the height from the midline to a peak), 2π/|B| ontal shift or phase shift, and D is the vertical shift that moves the midline up or down.
一般形式的余弦函数写作 y = A cos(B(x - C)) + D，这些参数控制着图像的形状与位置。|A| 是振幅（从中线到波峰的高度），2π/|B| 是周期（一个完整循环的长度），C 是水平位移或相位移，D 是垂直位移，使中线上移或下移。
by π), the graph is reflected vertically due to the negative A, and the midline is y = 4.
例如，在 y = -3 cos(½ (x + π)) + 4 中，振幅为 3，周期为 4π，相位移为 -π（向左 π），因 A 为负图像发生垂直反射，中线为 y = 4。
4. Key Properties: Domain, Range, and Even Nature | 关键性质：定义域、值域与偶函数特性
The domain of cos x is all real numbers, written as ℝ s(x) for all x. Graphically, this means the curve is symmetric with respect to the y-axis.
cos x 的定义域是所有实数，记作 ℝ，值域是闭区间 [-1, 1]。余弦是偶函数：对任意 x 都有 cos(-x) = cos(x)。从图像上看，这意味着曲线关于 y 轴对称。
perties include that cos x has maximum value 1 at x = 2πn and minimum -1 at x = π + 2πn (n ∈ ℤ s and differentiable everywhere on its domain, which is crucial for calculus applications.
其他重要性质包括：cos x 在 x = 2πn 处取得最大值 1，在 x = π + 2πn 处取得最小值 -1（n ∈ ℤ）。它在定义域上处处连续且可导，这对微积分应用很重要。
5. Transformations of the Cosine Function | 余弦函数的变换
By applying combinations of stretches, compressions, reflections, and translations, you can model a vast array of real-world situations. A vertical stretch by factor |A| changes the amplitude; if A is negative, the graph is reflected across the midline. A horizontal stretch by factor 1/|B| alters the period; if B is negative, the graph is reflected horizontally, though the even property often masks this.
通过组合拉伸、压缩、反射和平移，你可以对大量现实情景进行建模。以 |A| 为系数的垂直拉伸改变振幅；如果 A 为负，图像关于中线反射。以 1/|B| 为系数的水平拉伸改变周期；如果 B 为负，图像水平反射，但偶函数性质常常掩盖了这一点。
nto the standard form y = A cos(B(x - C)) + D first to correctly identify the phase shift.
由 C 给出的水平平移使图像向右（C > 0）或向左移动，而垂直平移 D 使正弦轴上下移动。务必先将函数写为标准形式 y = A cos(B(x - C)) + D，以正确识别相位移。
6. Solving Equations Involving Cosine | 求解含余弦的方程
To solve an equation like cos x = k, where k ∈ symmetry and periodicity, the general solutions are x = ±α + 2πn, where n is any integer.
要求解如 cos x = k 的方程，其中 k ∈ [-1, 1]，首先找出位于 [0, π] 的主值 α = arccos(k)。然后利用对称性和周期性，得到通解 x = ±α + 2πn，n 为任意整数。
cos(5π/3) = cos(π/3) due to symmetry). All solutions are x = π/3 + 2πn and x = 5π/3 + 2πn.
例如，若 cos x = ½，则 α = π/3。在 [0, 2π] 区间内的解为 x = π/3 和 x = 5π/3（因为 cos(5π/3) = cos(π/3)，由对称性得出）。所有解为 x = π/3 + 2πn 和 x = 5π/3 + 2πn。
he entire angle, then isolate x. Careful handling of the period and multiple cycles is essential to avoid missing solutions.
当自变量更复杂时，例如 cos(2x - π) = 0.5，首先对整体角度求解，再分离 x。仔细处理周期和多个循环对于避免漏解至关重要。
7. Cosine in Right-Angled Triangles | 直角三角形中的余弦
: cos θ = adjacent/hypotenuse. This is summarised in the SOHCAHTOA mnemonic with cosine as 'CAH'.
在直角三角形中，锐角 θ 的余弦定义为邻边长度与斜边长度之比：cos θ = 邻边/斜边。这在 SOHCAHTOA 记忆法中概括为余弦即 “CAH”。
h the ground, the distance from the wall to the base of the ladder is 5 × cos 60° = 2.5 m.
这一定义用于求未知边长或角度。例如，一把 5 m 长的梯子靠墙放置，与地面成 60° 角，则梯脚到墙的距离为 5 × cos 60° = 2.5 m。
8. The Law of Cosines | 余弦定理
2ab cos C. This formula can be rearranged to find an angle when all three sides are known.
余弦定理将余弦关系推广到任意三角形，而不仅仅是直角三角形。它指出，对于任意边长为 a、b、c 且对角分别为 A、B、C 的三角形，有 c² = a² + b² - 2ab cos C。当已知三边时，可重排该公式来求角度。
unique angle. It also reduces to the Pythagorean theorem when C = 90°, since cos 90° = 0.
它在含糊情形（SSA）中尤其有用，此时正弦定理可能给出两个可能的三角形，而余弦定理给出唯一的角度。当 C = 90° 时，由于 cos 90° = 0，它退化为勾股定理。
9. Derivatives and Integrals of Cosine | 余弦的导数与积分
u) is -sin(u) · du/dx. Thus, for example, d/dx [cos(3x²)] = -sin(3x²) · 6x = -6x sin(3x²).
对于 IB 微积分，cos x 关于 x 的导数是 -sin x。利用链式法则，cos(u) 的导数为 -sin(u) · du/dx。因此，例如 d/dx [cos(3x²)] = -sin(3x²) · 6x = -6x sin(3x²)。
more generally, ∫ cos(u) du = sin(u) + C, provided the substitution is managed correctly.
不定积分 ∫ cos x dx = sin x + C。对于定积分，这给出了余弦曲线下的有号面积。cos(kx) 的积分为 (1/k) sin(kx) + C，更一般地，∫ cos(u) du = sin(u) + C，前提是正确地处理换元。
10. Modelling Periodic Phenomena | 周期现象的建模
Cosine functions are ideal for modelling real-world cyclic data because they naturally oscillate between maximum and minimum values. Common applications include tidal height, sound waves, alternating current voltage, and average monthly temperatures.
余弦函数非常适合对现实世界的周期性数据进行建模，因为它们自然地在最大值和最小值之间波动。常见的应用包括潮汐高度、声波、交流电压以及月平均气温。
model has the form y = A cos(B(t - C)) + D, where t represents time. For instance, a city' 6 (t - 1)) + 25, predicting a maximum of 40°C in month 1 and a minimum of 10°C in month 7.
典型的模型形式为 y = A cos(B(t - C)) + D，其中 t 代表时间。例如，某城市全年的气温可以建模为 T(t) = -15 cos(π/6 (t - 1)) + 25，预测第1个月最高温 40°C，第7个月最低温 10°C。
When fitting a cosine model, identify the amplitude from the difference between maximum and minimum values, the period from the length of one cycle, the vertical shift as the midline, and the phase shift from the timing of a maximum or minimum.
在拟合余弦模型时，根据最大值与最小值的差值确定振幅，根据一个循环的长度确定周期，垂直位移为中线的位置，相位移根据最大值或最小值出现的时间点确定。
11. The Inverse Cosine Function | 反余弦函数
in is [-1, 1], and its graph is the reflection of the restricted cosine curve y = cos x, x ∈ [0, π] across the line y = x.
反余弦函数写作 arccos x 或 cos⁻¹ x，返回 [0, π] 中余弦值为 x 的唯一角度。它的定义域是 [-1, 1]，其图像是限定在 [0, π] 上的余弦曲线 y = cos x 关于直线 y = x 的反射。
= π/3 gives one solution; the other solutions come from the symmetry properties of cosine.
不要将 cos⁻¹ x 与 (cos x)⁻¹ = 1/cos x = sec x 混淆，这一点很重要。在解方程时，arccos(0.5) = π/3 提供一个解；其余解由余弦的对称性得出。
12. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧
ative solutions when solving cos x = k; remember that ± α both give the same cosine value.
许多学生忘记将计算器设置为正确的模式（弧度或角度），导致完全错误的答案。务必检查题目的背景：如果问题涉及 π，请使用弧度模式。另一个常见错误是求解 cos x = k 时忽略负解；要记住 ±α 给出相同的余弦值。
B to correctly find the phase shift as y = A cos(B(x - C)) + D, not y = A cos(Bx - C) + D.
混淆振幅与垂直位移也很常见：振幅衡量中线到波峰的距离，而不是总高度。处理变换时，请务必提取 B 以正确找出相位移，形式为 y = A cos(B(x - C)) + D，而非 y = A cos(Bx - C) + D。
During exams, quickly sketching a unit circle or a rough graph can verify solutions and prevent sign mistakes. Remember that cosine is positive in quadrants I and IV (angles between 0 and π/2, and 3π/2 to 2π), and negative in quadrants II and III.
考试中，快速画出单位圆或草图可以验证解并防止符号错误。记住余弦在第一象限和第四象限（0 到 π/2，以及 3π/2 到 2π 的角）为正，在第二和第三象限为负。
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