📚 Exercise 10B.1: Trigonometric Functions and Equations | 练习10B.1：三角函数与方程
In IB Mathematics, Exercise 10B.1 typically introduces the core concepts of trigonometric functions, including radian measure, the unit circle, and the graphical behaviour of sine and cosine. Mastering these fundamentals is essential for solving equations and modelling periodic phenomena in both Analysis and Approaches and Applications and Interpretation courses.
在IB数学中，练习10B.1通常介绍三角函数的核心概念，包括弧度制、单位圆以及正弦和余弦的图像特征。掌握这些基础知识对于在分析与方法以及应用与解释课程中解方程和模拟周期现象至关重要。
1. Radian Measure | 弧度制
A radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. One full revolution corresponds to 2π radians, establishing a natural link between angle measure and arc length.
弧度定义为圆上弧长等于半径时所对的圆心角。一整圈对应 2π 弧度，从而在角度度量与弧长之间建立了自然的联系。
m degrees to radians, multiply by π/180. To go from radians to degrees, multiply by 180/π.
转换关系非常简单：π rad = 180°。从度转换为弧度乘以 π/180；从弧度转换为度则乘以 180/π。
d, 60° = π/3 rad and 90° = π/2 rad. IB papers often require answers exactly in terms of π.
为加快IB考试中的解题速度，常见角度的转换值得熟记。例如，30° = π/6 rad，45° = π/4 rad，60° = π/3 rad，90° = π/2 rad。IB试卷常要求用含π的精确值作答。
2. The Unit Circle | 单位圆
The unit circle is a circle of radius 1 centred at the origin (0,0). Any point on the circumference can be expressed as (cos θ, sin θ), where θ is the angle measured anticlockwise from the positive x-axis.
单位圆是以原点 (0,0) 为圆心、半径为1的圆。圆周上任意一点都可以表示为 (cos θ, sin θ)，其中 θ 是从正 x 轴逆时针测量的角度。
This geometric representation extends the trigonometric functions to all real numbers. Angles are no longer limited to acute triangles; we can define sin θ and cos θ for any θ, including negative angles and angles greater than 360° (or 2π rad).
这种几何表示将三角函数扩展到所有实数。角度不再局限于锐角三角形；我们可以定义任意 θ 的 sin θ 和 cos θ，包括负角以及大于 360°（或 2π 弧度）的角。
the unit circle at angle -π/2 has coordinates (0,-1), so cos(-π/2) = 0 and sin(-π/2) = -1.
顺时针旋转给出负角。例如，单位圆上角度为 -π/2 的点坐标为 (0,-1)，因此 cos(-π/2) = 0 且 sin(-π/2) = -1。
3. Defining Sine and Cosine | 正弦与余弦的定义
Using the unit circle, cos θ is the x-coordinate and sin θ is the y-coordinate of the terminal point of the arc of length θ starting from (1,0). This definition works for any real θ and makes the periodic nature of these functions clear.
利用单位圆，cos θ 是从 (1,0) 出发、弧长为 θ 的终点之 x 坐标，sin θ 是其 y 坐标。此定义适用于任意实数 θ，并清晰地呈现了这些函数的周期性。
. These symmetry properties follow directly from the reflection of points across the axes.
正弦函数是奇函数：sin(-θ) = -sin θ；而余弦函数是偶函数：cos(-θ) = cos θ。这些对称性质直接源于点关于坐标轴的反射。
The Pythagorean identity derived from the unit circle is central to simplifying expressions:
sin²θ + cos²θ = 1
由单位圆导出的毕达哥拉斯恒等式是化简表达式的核心：
sin²θ + cos²θ = 1
4. Graphs of sin x and cos x | 正弦与余弦的图像
ns to 0 at x = π, drops to a minimum of -1 at x = 3π/2, and completes one cycle at x = 2π.
y = sin x 的图像是一条光滑的波形，周期为 2π。它从原点 (0,0) 出发，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，并在 x = 2π 处完成一个周期。
left by π/2. It starts at its maximum (0,1), giving it the property cos x = sin(x + π/2).
y = cos x 的图像形状相同，但向左平移了 π/2。它从最大值 (0,1) 出发，因此有 cos x = sin(x + π/2) 的性质。
Both graphs have amplitude 1 and period 2π. Understanding these standard curves is vital for predicting the behaviour of transformed trigonometric functions.
两幅图像的振幅均为 1，周期均为 2π。理解这些标准曲线对于预测变换后的三角函数行为至关重要。
5. The Tangent Function | 正切函数
rresponds to the length of the segment of the tangent line to the unit circle at (1,0) from the x-axis to the terminal ray of the angle.
正切函数定义为 tan θ = sin θ / cos θ。几何上，它对应从 x 轴到角终边的单位圆在 (1,0) 处的切线段长度。
y = tan x. The tangent function has period π, repeating twice as often as sine and cosine.
由于 cos θ = 0 当 θ = π/2 + kπ（k 为整数）时，这些值就是 y = tan x 图像的垂直渐近线。正切函数的周期为 π，其重复频率是正弦和余弦的两倍。
pproaches π/2 from the left, and decreases towards -∞ as x approaches -π/2 from the right.
tan x 的图像穿过原点，当 x 从左侧趋近 π/2 时增加至 +∞，当 x 从右侧趋近 -π/2 时减小至 -∞。
6. Exact Values for Common Angles | 常见角度的精值
IB candidates must recall exact trigonometric values without a calculator for angles 0, π/6, π/4, π/3, π/2 and their multiples. These are derived from the special triangles: the 45°-45°-90° triangle (sides 1, 1, √2) and the 30°-60°-90° triangle (sides 1, √3, 2).
IB 考生必须在不使用计算器的情况下记住 0、π/6、π/4、π/3、π/2 及其倍数角的三角函数精值。这些值可从特殊三角形推导得出：45°-45°-90° 三角形（边长 1, 1, √2）和 30°-60°-90° 三角形（边长 1, √3, 2）。
| θ (rad) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| π/6 | ½ | √3/2 | 1/√3 |
| π/4 | √2/2 | √2/2 | 1 |
| π/3 | √3/2 | ½ | √3 |
| π/2 | 1 | 0 | undefined |
Learning to rationalise denominators, such as writing 1/√2 as √2/2, is expected in final answers. These exact values are the building blocks for solving equations without a GDC.
习得分母有理化，例如将 1/√2 写作 √2/2，是最终答案中的要求。这些精值是无需图形计算器解方程的基石。
7. Solving Basic Trigonometric Equations | 解基本三角方程
y given a domain such as 0 ≤ x ≤ 2π, and you must find all solutions within that interval.
像 sin x = ½ 这样的简单方程因周期性而有无穷多解。在 IB 题目中，通常会给定一个区间，如 0 ≤ x ≤ 2π，你必须找出该区间内的所有解。
= π - π/6 = 5π/6. A general solution can be written using the patterns of the sine curve.
sin x = ½ 的主解为 x = π/6。利用正弦图像的对称性，在 [0, 2π] 内的第二个解是 x = π - π/6 = 5π/6。通解可利用正弦曲线的规律写出。
he solutions repeat every π and there is exactly one solution in any interval of length π.
对于 cos x = a，若 -1 ≤ a ≤ 1 且 a ≠ 1, -1，一个周期内的两个解为 θ 和 2π - θ。对于 tan x = a，解每隔 π 重复一次，在任意长度为 π 的区间内恰好有一解。
8. Transformations of Trigonometric Graphs | 三角函数图像的变换
A general sine function can be written as y = A sin(B(x - C)) + D. Here, |A| is the amplitude, 2π/|B| s the period, C is the horizontal shift, and D is the vertical shift (the principal axis).
一般正弦函数可写作 y = A sin(B(x - C)) + D。其中 |A| 为振幅，2π/|B| 为周期，C 为水平平移，D 为垂直平移（主轴线）。
When sketching, it helps to map the key points of the basic curve through the transformations. Start with the vertical scaling and reflection (A), then horizontal scaling and shift (B and C), and finally the vertical translation (D).
在绘制草图时，将基本图像的关键点通过变换进行映射会很有帮助。先从垂直缩放和反射 (A) 开始，然后是水平缩放和平移 (B 和 C)，最后是垂直平移 (D)。
Understanding transformations is essential for modelling periodic data, such as tides or temperatures, where a sinusoidal function must be fitted to real-world measurements.
理解图像变换对于模拟周期性数据（例如潮汐或温度）至关重要，因为在这些情况下需要将正弦函数拟合到实际测量数据中。
9. Double Angle Formulas | 倍角公式
The double angle identities extend the toolkit for manipulating trigonometric expressions. The three most useful forms are:
倍角恒等式扩展了处理三角表达式的工具包。三种最常用的形式如下：
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ - sin²θ = 2 cos²θ - 1 = 1 - 2 sin²θ
tan 2θ = 2 tan θ / (1 - tan²θ)
Which version of cos 2θ to use depends on the form of the expression you are trying to simplify. For instance, 1 - 2 sin²θ is helpful when solving equations that only involve sine.
使用哪种形式的 cos 2θ 取决于你要化简的表达式的形式。例如，当解只涉及正弦的方程时，1 - 2 sin²θ 就很方便。
in solving equations such as sin 2x = sin x, where rewriting sin 2x allows factorisation.
这些公式经常出现在 IB 证明题以及求解方程（如 sin 2x = sin x）中，此时改写 sin 2x 可以实现因式分解。
10. Solving Equations Using Identities | 利用恒等式解方程
ype. A typical approach is to use sin²x + cos²x = 1 to replace, say, cos²x with 1 - sin²x.
当方程中包含多个三角函数时，可利用恒等式将其化为单一类型。一个典型的方法是使用 sin²x + cos²x = 1，例如将 cos²x 替换为 1 - sin²x。
0, rewrite sin²x as 1 - cos²x to get a quadratic in cos x. You must also check for extraneous solutions that might arise from squaring steps.
例如，要解 2 sin²x + 3 cos x = 0，可将 sin²x 改写为 1 - cos²x，从而得到关于 cos x 的二次方程。同时，必须检验因平方步骤可能产生的增根。
In IB exams, the domain of the variable must be respected. After finding general solutions, identify the specific values that lie within the requested interval, often writing them as exact multiples of π.
在 IB 考试中，必须遵守变量的定义域。找到通解后，识别出位于所求区间内的具体值，通常写作 π 的精确倍数。
11. Modelling with Trigonometric Functions | 用三角函数建模
e t is time in hours. The amplitude A represents the maximum variation from the mean water level.
许多 IB 实际应用题涉及将正弦或余弦模型拟合到数据中。典型的潮汐高度函数可能是 h(t) = A sin(B(t - C)) + D，其中 t 是时间（小时）。振幅 A 代表相对于平均水位的最大变化。
is 12, giving B = 2π/12 = π/6. The vertical shift D is the average value of the function.
周期由现象的重复频率决定。对于每 12 小时重复一次的潮汐，周期为 12，因此 B = 2π/12 = π/6。垂直平移 D 是函数的平均值。
Once the model is built, it can be used to predict values at specific times or to find when the height exceeds a certain level, requiring you to solve a trigonometric inequality.
模型建立后，可用于预测特定时间的值，或找出高度超过某一水平的时刻，这就需要求解三角不等式。
12. Summary of Key Concepts | 核心概念总结
Exercise 10B.1 builds fluency with radian thinking, the unit circle, and the fundamental graphs. These are not isolated topics; they form the backbone of calculus with trigonometric functions, complex numbers, and vectors in further IB studies.
练习 10B.1 旨在培养弧度思维、单位圆和基本图像方面的熟练度。这些并非孤立的主题，它们构成了 IB 后续学习中三角函数的微积分、复数以及向量等内容的支柱。
Practice consistently with equation solving in given intervals and with graph transformations. By committing the exact value table and key identities to memory, you will approach both examination questions and internal assessments with confidence.
持续练习在给定区间内解方程以及图像变换。熟记精值表和关键恒等式后，你将能自信地应对考试题目和内部评估。
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