Mixed Exercise 10: Trigonometric Identities and Equations | 第10章混合练习：三角恒等式与方程

📚 Mixed Exercise 10: Trigonometric Identities and Equations | 第10章混合练习：三角恒等式与方程

Mixed Exercise 10 brings together the key techniques from the trigonometric identities and equations chapter. You need to recall exact values, apply the two core identities, and solve equations in a given interval without losing solutions. This article breaks the exercise down into focused revision points.

第10章混合练习整合了三角恒等式与方程一章的核心技巧。你需要熟记特殊角的精确值，应用两个基本恒等式，并在给定区间内解方程而不丢失解。本文把该练习拆解为几个重点复习模块。

The exercise is not testing isolated facts. It tests whether you can look at a messy expression, decide which identity will simplify it, and then carry the algebra through to a complete solution set. That is exactly why it is called a mixed exercise: the method is not given to you in the question.

这套练习考查的不是孤立的知识点，而是你能否面对一个复杂的表达式，判断该用哪个恒等式来化简，并把代数运算完整推进到最终解集。这正是它被称为混合练习的原因：题目不会直接告诉你方法。

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1. The Big Picture: What Mixed Exercise 10 Tests | 全局概览：混合练习10考查什么

Mixed Exercise 10 usually contains two families of question: prove an identity, and solve an equation in a stated interval. The examiner wants to see whether you choose the correct identity, rewrite an expression cleanly, and find every solution inside that interval.

混合练习10通常包含两类题：证明恒等式，以及在给定区间内解方程。考官想检验你是否会选择正确的恒等式、能否干净地重写表达式，并在该区间内找出全部解。

l first, because it decides how many solutions you must give and whether a solution just outside the interval should be rejected.

区间通常是 0° ≤ θ ≤ 360° 或 0 ≤ θ ≤ 2π，也有题目用 −180° ≤ θ ≤ 180°。一定先读区间，因为它决定你需要给出多少个解，以及某个刚好落在区间外的解是否应当舍去。

Questions often combine exact values from the 30°, 45°, 60° family with sign rules from the four quadrants. The exercise also includes equations that become quadratics in sin θ or cos θ, so algebraic factorising is as important as trigonometric knowledge.

题目常把 30°、45°、60° 一族的精确值与四个象限的正负号规则结合起来考查。练习中还包含可化为 sin θ 或 cos θ 二次型的方程，因此代数因式分解与三角知识同样重要。

− 1 = 0 for 0° ≤ θ ≤ 360°. Both tasks reward clear, ordered working rather than guesswork.

典型题目可能要求你证明某个表达式等于 tan θ，或者求 2 sin²θ + sin θ − 1 = 0 在 0° ≤ θ ≤ 360° 内的解。这两类任务都奖励清晰、有序的书写，而不是猜测。

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2. The Two Core Identities You Must Quote | 必须引用的两个核心恒等式

Everything in this chapter is built on two identities. The first is the Pythagorean identity, true for every value of θ. The second defines tan θ in terms of sine and cosine, and it is valid whenever cos θ ≠ 0.

本章的一切都建立在两个恒等式之上。第一个是毕达哥拉斯恒等式，对任意 θ 都成立。第二个用正弦和余弦定义 tan θ，只要 cos θ ≠ 0 就成立。

sin²θ + cos²θ = 1

tan θ = sin θ ÷ cos θ, cos θ ≠ 0

− cos²θ and cos²θ = 1 − sin²θ. Choosing the right rearrangement is often the whole method.

由第一个恒等式可以立刻写出两个在混合练习中反复出现的变形：sin²θ = 1 − cos²θ 与 cos²θ = 1 − sin²θ。选对变形往往就是整道题的方法所在。

When a question mixes sin²θ and cos θ, replace sin²θ with 1 − cos²θ. When it mixes cos²θ and sin θ, replace cos²θ with 1 − sin²θ. The goal is always to leave one trigonometric function in one variable.

当题目把 sin²θ 与 cos θ 混在一起时，把 sin²θ 换成 1 − cos²θ。当题目把 cos²θ 与 sin θ 混在一起时，把 cos²θ 换成 1 − sin²θ。目标始终是只留下一个三角函数、一个变量。

For proof questions, quote the identity by name or by formula before you use it. Edexcel mark schemes frequently award a method mark for a correctly stated identity, even if later algebra slips.

证明题中，使用恒等式之前要先写出它的名称或公式。Edexcel 评分标准常常对正确写出的恒等式给方法分，即使后面的代数出现失误。

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3. Exact Values for 30°, 45°, 60° | 30°、45°、60°的精确值

Exact values are non-negotiable in Mixed Exercise 10. If a question asks for a solution in surd form, a decimal answer from your calculator will lose the accuracy mark. Learn the table below until you can reproduce it in under a minute.

精确值在混合练习10中不可回避。如果题目要求以根式形式给出解，用计算器得到的近似小数会丢失精度分。把下表练到一分钟内默写出来。

θ	0°	30°	45°	60°	90°
sin θ	0	½	√2 ÷ 2	√3 ÷ 2	1
cos θ	1	√3 ÷ 2	√2 ÷ 2	½	0
tan θ	0	√3 ÷ 3	1	√3	未定义

Two memory aids help. First, the sine row increases from 0 to 1 while the cosine row is the same sequence reversed. Second, every value can be written over 2 except tan 45° and tan 60°, which follow from sin ÷ cos.

两个记忆法很有帮助。第一，正弦行从 0 递增到 1，而余弦行是同一串数反过来。第二，除 tan 45° 与 tan 60° 由 sin ÷ cos 推出外，其余值都可以写成某个数除以 2。

; only the way you write the angle changes. Keep your calculator in the mode the question uses.

用弧度制时，30° = π ÷ 6，45° = π ÷ 4，60° = π ÷ 3。精确值本身不变，改变的只是角的写法。务必让计算器处于题目要求的模式。

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4. Signs in the Four Quadrants | 四个象限的正负号

Trigonometric equations usually have more than one solution in the interval, and those extra solutions come from the symmetry of the graph. The sign of each function in each quadrant tells you which symmetries to use.

三角方程在区间内通常不止一个解，这些额外的解来自函数图像的对称性。每个函数在各象限的正负号告诉你该使用哪些对称关系。

象限	cos θ	sin θ	tan θ
I (0°–90°)	+	+	+
II (90°–180°)	−	+	−
III (180°–270°)	−	−	+
IV (270°–360°)	+	−	−

alue, or forgetting that a negative sine places the angle in the third or fourth quadrant.

开始解题前先在草稿纸顶部写下各象限的正负号。这样可以避免经典错误：把 cos θ = −0.5 当成正值处理，或者忘记负的正弦对应的角在第三或第四象限。

The related-angle formulas follow directly from these signs. You do not need to memorise them separately if you can see the symmetry on the unit circle.

相关角公式直接来自这些正负号。只要能在单位圆上看出对称关系，就不必单独死记这些公式。

sin(180° − θ) = sin θ   cos(180° − θ) = −cos θ

sin(180° + θ) = −sin θ   cos(180° + θ) = −cos θ

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sin(360° − θ) = −sin θ   cos(360° − θ) = cos θ

− θ) = −cos θ, and cos(2π − θ) = cos θ. The pattern is identical; only the labels change.

用弧度制表达同样成立：sin(π − θ) = sin θ，cos(π − θ) = −cos θ，cos(2π − θ) = cos θ。规律完全相同，只是标记改变了。

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5. CAST Diagrams and Reference Angles | CAST 图与参考角

A CAST diagram is a compact way to remember which function is positive in each quadrant. Starting in the fourth quadrant and moving anticlockwise, the letters read C, A, S, T: Cos positive, All positive, Sin positive, Tan positive.

CAST 图是记住各象限哪个函数为正的紧凑方法。从第四象限开始逆时针移动，字母依次读作 C、A、S、T：余弦为正、全部为正、正弦为正、正切为正。

The reference angle, sometimes called the acute angle, is the angle between the terminal side of θ and the horizontal axis. It is always between 0° and 90°, and it is the angle whose exact value you look up.

参考角也称锐角，是 θ 的终边与水平轴之间的夹角。它总在 0° 与 90° 之间，也是你查精确值时所使用的那个角。

Method for an equation such as sin θ = −√3 d and fourth quadrants. The solutions are therefore 180° + 60° = 240° and 360° − 60° = 300°.

以 sin θ = −√3 ÷ 2 在 0° ≤ θ ≤ 360° 内为例：先忽略负号求出参考角，得到 60°。再用 CAST 图看出正弦在第三、第四象限为负。因此解为 180° + 60° = 240° 与 360° − 60° = 300°。

α. For tangent, the period is 180°, so once you have one solution α, the next is α + 180°.

对余弦而言，对称轴是水平轴，所以负的余弦给出的解在第二、第三象限：θ = 180° − α 与 θ = 180° + α。对正切而言，周期是 180°，所以得到一个解 α 后，下一个解是 α + 180°。

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6. Proving Identities: Strategy and Layout | 证明恒等式：策略与书写

An identity proof asks you to show that two expressions are equal for every allowed value of θ. The golden rule is to work on one side only, usually the more complicated side, and transform it into the other side.

恒等式证明要求你展示两个表达式在所有允许的 θ 值下都相等。黄金法则是只处理一边，通常选择更复杂的一边，把它变形为另一边。

Never move terms across the equals sign in a proof. That would be solving an equation, not proving an identity. Write a clear heading such as left-hand side, then show a chain of equal expressions, and finish with the statement that it equals the right-hand side.

证明中绝不要把项移过等号。那是在解方程，而不是证明恒等式。清楚地写出左边这一标题，然后展示一串相等的表达式，最后说明它等于右边。

; combining fractions over a common denominator; and using sin²θ + cos²θ = 1 to eliminate squared terms.

混合练习10中常见的策略包括：把 tan θ 换成 sin θ ÷ cos θ；分子分母同乘共轭式；把分数通分到同一分母；以及用 sin²θ + cos²θ = 1 消去平方项。

If you get stuck, write everything in terms of sin θ and cos θ. Very few identities in this chapter resist that first move, because the two core identities are the only tools you need most of the time.

如果卡住了，就把所有东西都写成 sin θ 与 cos θ。本章中很少有恒等式能抵抗这一步，因为大多数时候你需要的工具只有那两个核心恒等式。

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7. Solving Basic Equations in a Given Interval | 在给定区间内解基本方程

mber. The method has three steps: find the principal value or reference angle, use symmetry to find all angles in one full turn, then select those inside the stated interval.

基本方程的形式是 sin θ = k、cos θ = k 或 tan θ = k，其中 k 是已知数。方法分三步：求出主值或参考角；利用对称性求出一整圈内的所有角；再选出落在给定区间内的那些角。

0° to each solution and check whether the new values still lie inside the interval. Students lose marks here more often than anywhere else in the exercise.

如果区间大于 360°，例如 0° ≤ θ ≤ 720°，你必须给每个解加上 360°，并检查新值是否仍在区间内。学生在这里丢分比在练习中其他任何地方都多。

The general solutions, useful for checking, are written as follows.

便于检验的一般解可以写成下面这样。

sin θ = k → θ = α + 360°n or θ = 180° − α + 360°n

cos θ = k → θ = ±α + 360°n

tan θ = k → θ = α + 180°n

Here α is the reference angle and n is any integer. In radians, replace 360° by 2π and 180° by π. Always finish by listing the solutions in ascending order, because that is what the mark scheme expects.

其中 α 是参考角，n 是任意整数。用弧度制时，把 360° 换成 2π，把 180° 换成 π。最后一定要按从小到大的顺序列出所有解，这是评分标准所期望的形式。

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8. Equations That Become Quadratics | 化为二次型的方程

Many Mixed Exercise 10 questions hide a quadratic. The trick is to notice that the equation contains a squared trigonometric function and the same function to the first power, with no other functions present.

混合练习10中许多题目隐藏着一个二次方程。诀窍是注意到方程中同时出现了某个三角函数的平方项和一次项，而且没有其他函数出现。

e variable, factorise as (2 sin θ − 1)(sin θ + 1) = 0, and split into two basic equations.

例题：求 2 sin²θ + sin θ − 1 = 0 在 0° ≤ θ ≤ 360° 内的解。把 sin θ 看成一个变量，因式分解为 (2 sin θ − 1)(sin θ + 1) = 0，再拆成两个基本方程。

θ + 1 = 0 we get sin θ = −1, giving θ = 270°. The full solution set is 30°, 150° and 270°.

由 2 sin θ − 1 = 0 得 sin θ = ½，于是 θ = 30° 与 θ = 150°。由 sin θ + 1 = 0 得 sin θ = −1，于是 θ = 270°。完整解集为 30°、150°、270°。

a. Always check whether a root is outside the range −1 ≤ sin θ ≤ 1 or −1 ≤ cos θ ≤ 1; such

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