📚 Chapter Review 9: Trigonometric Ratios | 第九章复习：三角比
In the Edexcel AS Pure Mathematics course, Chapter 9 brings together the key tools of triangle geometry and periodic functions. This review consolidates the cosine rule, sine rule, area of a triangle, graphs of sine, cosine and tangent, exact values, equation solving and graph transformations. Mastering these will give you confidence in both calculator and non-calculator exam questions.
在 Edexcel AS 纯数学课程中，第九章综合了三角形几何与周期函数的核心工具。本次复习将余弦定理、正弦定理、三角形面积、正弦余弦正切图像、精确值、解方程以及图像变换等内容串联起来。熟练掌握这些内容将使你在可使用计算器和不可使用计算器的考题中都充满信心。
1. The Cosine Rule | 余弦定理
The cosine rule relates the three sides of a triangle to the cosine of one of its angles. It is especially useful when you know two sides and the included angle, or when you know all three sides and need an angle.
余弦定理将三角形的三条边与其中一个角的余弦联系起来。当已知两条边及其夹角，或已知三条边求角时，余弦定理尤为有用。
a² = b² + c² − 2bc cos A
e angle C. You can rearrange the formula to find an angle: cos A = (b² + c² − a²) / (2bc).
对于通常标记的三角形，边 a 对角 A，边 b 对角 B，边 c 对角 C。你可以将公式变形来求角：cos A = (b² + c² − a²) / (2bc)。
Always check that the angle you are working with is the one opposite the side you are solving for. Remember to take the square root carefully and consider if the triangle could have an obtuse angle.
始终确保所处理的角是所求边的对角。牢记平方根要仔细开方，并考虑三角形是否存在钝角的可能性。
2. The Sine Rule | 正弦定理
The sine rule is a powerful tool for non‑right‑angled triangles. It works when you know two angles and a side, or two sides and a non‑included angle.
正弦定理是处理非直角三角形的强大工具。当你已知两个角与一条边，或已知两条边及一个非夹角时，可以使用它。
a / sin A = b / sin B = c / sin C
an angle, be mindful of the ambiguous case: if you use the sine rule to find an angle, there may be both an acute and an obtuse solution. Always check whether the angle could be 180° minus the acute value.
你还可以使用倒数形式 sin A / a = sin B / b = sin C / c。求角时要注意模糊情形：如果用正弦定理求角，可能存在锐角和钝角两个解。务必检查该角是否可能是 180° 减去锐角值。
In the ambiguous case, sketch a diagram and check if the given information really allows a second triangle.
在模糊情形中，画出示意图，确认给定的条件是否确实允许第二个三角形存在。
3. Area of a Triangle | 三角形面积
When you know two sides and the angle between them, the area of a triangle is given by:
当你已知两条边及其夹角时，三角形面积公式为：
Area = ½ ab sin C
This formula also works for any combination of sides and their included angle, such as ½ bc sin A or ½ ac sin B. It does not require the height to be perpendicular; the sine function automatically accounts for the vertical component.
该公式同样适用于其他边与其夹角的组合，例如 ½ bc sin A 或 ½ ac sin B。它不需要垂线高度；正弦函数自动计入了垂直分量。
This area formula is particularly useful when you are dealing with irregular shapes that can be split into triangles.
在处理可分割为三角形的不规则形状时，该面积公式尤其有用。
4. Solving Triangle Problems | 解三角形问题
Many exam questions ask you to find unknown sides, angles or areas by combining the cosine rule, sine rule and area formula. The key is to decide which rule fits the data you have.
许多考试题要求你综合运用余弦定理、正弦定理和面积公式来求未知的边、角或面积。关键在于根据已知数据选择合适的定理。
A typical strategy: label the triangle clearly, write down what you know and what you need, then check whether the cosine rule or sine rule applies first. After finding one missing angle or side, you can then use basic angle sums or another rule to finish the solution.
典型策略：清晰地标记三角形，写下已知量和所求量，然后判断先使用余弦定理还是正弦定理。在求出一个未知角或边之后，你可以利用基本角度和或者其他定理来完成解答。
Watch out for units – sides in cm, angles in degrees. Always give your answer to a sensible degree of accuracy unless told otherwise.
注意单位——边长为 cm，角度为度。除非题目另有规定，答案要保留合适的精度。
5. Graphs of Sine, Cosine and Tangent | 正弦、余弦和正切函数的图形
n x are periodic and have distinctive shapes. You need to know their key features: domain, range, period, symmetry and where they cross the axes.
函数 y = sin x、y = cos x 和 y = tan x 的图像是周期性的，形状各具特色。你需要掌握它们的主要特征：定义域、值域、周期、对称性以及与坐标轴的交点。
sin x: passes through (0,0), period 360°, range [−1, 1], maximum at 90°, minimum at 270°.
y = sin x：经过 (0,0)，周期 360°，值域 [−1, 1]，在 90° 处有最大值，在 270° 处有最小值。
passes through (0,1), period 360°, range [−1, 1], maximum at 0° and 360°, minimum at 180°.
y = cos x：经过 (0,1)，周期 360°，值域 [−1, 1]，在 0° 和 360° 处取最大值，在 180° 处取最小值。
y = tan x: period 180°, asymptotes at x = 90°, 270°…, passes through (0,0), range all real numbers.
y = tan x：周期 180°，渐近线在 x = 90°, 270°…，经过 (0,0)，值域为所有实数。
Being able to sketch these graphs quickly will help you visualise trigonometric equations and transformations.
能够快速画出这些草图，将帮助你直观理解三角方程和图像变换。
6. Exact Trigonometric Values | 精确三角值
You must memorise the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°. These come from two special triangles: the equilateral triangle (side 2, half to get 60° and 30°) and the right isosceles triangle (legs 1, hypotenuse √2).
你必须记住 0°、30°、45°、60° 和 90° 的正弦、余弦和正切的精确值。它们来自两个特殊三角形：等边三角形（边长 2，分割得到 60° 和 30°）和等腰直角三角形（直角边 1，斜边 √2）。
| Angle / 角度 | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3 / 2 | 1 / √3 |
| 45° | 1 / √2 | 1 / √2 | 1 |
| 60° | √3 / 2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
You can often use these exact values to give simplified surd answers, which avoid rounding errors and earn full marks on non‑calculator papers.
你通常可以使用这些精确值给出简化的根号答案，避免舍入误差，并在非计算器试卷上获得满分。
7. Solving Trigonometric Equations | 解三角方程
the exam. The approach is to find the principal value, then use the symmetry of the graphs or the CAST diagram to find all solutions within the given interval.
诸如 sin x = 0.5、cos 2θ = -0.4 或 tan (x − 30°) = 1 这样的方程贯穿整个考试。解题方法是求出主值，再利用图像的对称性或 CAST 图找出给定区间内的所有解。
For sine and cosine: the second solution is usually 180° minus or plus the principal value, depending on the quadrant. For tangent: add or subtract 180° because the period is 180°.
对于正弦和余弦：第二个解通常是 180° 减去或加上主值，取决于象限。对于正切：周期为 180°，可以加上或减去 180°。
.g. if you need to solve sin (2x) = 0.5 for 0° ≤ x ≤ 360°, work with 0° ≤ 2x ≤ 720° first.
始终先确定括号内角度的范围：例如，若要在 0° ≤ x ≤ 360° 上解 sin (2x) = 0.5，则先在 0° ≤ 2x ≤ 720° 上求解。
8. Transforming Trigonometric Graphs | 三角图形的变换
slated. The general forms are y = a sin (bx + c) + d and similarly for cosine and tangent.
三角函数的图像可以进行拉伸、反射和平移。一般形式为 y = a sin (bx + c) + d，余弦和正切类似。
a changes the vertical stretch (amplitude).
a 改变垂直拉伸（振幅）。
b affects the period: period = 360° / |b| for sine and cosine, and 180° / |b| for tangent.
b 影响周期：对正弦和余弦，周期 = 360° / |b|；对正切，周期 = 180° / |b|。
c causes a horizontal translation (phase shift) of −c / b.
c 导致水平平移（相移），平移量为 −c / b。
d gives a vertical translation.
d 给出垂直平移。
When sketching a transformed graph, start with the basic shape, apply stretches first, then translations. Label the intersections with the axes and the positions of maximum and minimum points clearly.
在画变换图像时，从基本形状开始，先进行拉伸变换，然后是平移。清楚地标出与坐标轴的交点以及最大值和最小值的位置。
9. Mixed Practice and Exam Tips | 综合练习与考试技巧
Here are some final points to help you on exam day:
以下是一些在考试中有帮助的最后要点：
Always write down the formula you are using, even if it is given in the formula booklet. It shows your reasoning and can earn method marks.
一定要写下你使用的公式，即使公式表里提供了。这能展示你的推理过程，并能获得方法分。
Check if your calculator is in degree mode. A common mistake is to leave it in radian mode when working with degrees.
检查计算器是否处于度数模式。一个常见错误是在处理度数时将其留在了弧度模式。
Draw a sketch triangle for sine and cosine rule problems. It will prevent you mixing up sides and angles.
对于正弦和余弦定理问题，画出三角形草图，这能避免你混淆边与角。
When solving equations, list all solutions systematically and then check they lie within the required interval.
在解方程时，系统列出所有解，然后检查它们是否在要求的区间内。
With exact values, never replace a surd with a decimal unless the question specifically asks for it.
对于精确值，除非题目明确要求，否则不要用小数代替根号形式。
Practice as many mixed exercises as you can – Chapter Review 9 in the official textbook is an excellent starting point.
尽可能多地练习混合题——官方课本第九章的“Chapter Review 9”是一个极佳的起点。
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