📚 IB Math: Core Trigonometric Concepts and Exam Tips | IB数学：三角函数核心考点
Trigonometry is one of the most important and frequently tested topics in IB Mathematics. It appears in both Analysis and Approaches (AA) and Applications and Interpretation (AI), spanning geometry, algebra, functions, and calculus. Mastery of trigonometric core concepts and problem-solving techniques is essential for exam success.
三角函数是IB数学中最重要且最常考的主题之一。在分析与方法（AA）与应用与解释（AI）中都会出现，覆盖几何、代数、函数和微积分。掌握三角函数核心概念与解题技巧，是取得高分的关键。
1. Radian Measure and the Unit Circle | 弧度制与单位圆
Radian measure is essential in IB Mathematics. Angles measured in radians are defined by the arc length relative to the radius. A full circle is 2π radians, which equals 360°.
弧度制在IB数学中非常关键。以弧度为单位的角度由弧长与半径之比定义。一个完整圆周为2π弧度，等于360°。
On the unit circle, the point corresponding to angle θ has coordinates (cosθ, sinθ). The tangent is defined as tanθ = sinθcosθ.
在单位圆上，与角θ对应的点坐标为(cosθ, sinθ)。正切定义为tanθ = sinθ / cosθ。
The arc length formula in radians is s = rθ, and the sector area is A = 12r2θ. These formulas are only valid when θ is in radians.
弧度制下的弧长公式为s = rθ，扇形面积公式为A = 12r2θ。这些公式仅在θ以弧度为单位时成立。
A common conversion is: π rad = 180°. To convert from degrees to radians, multiply by π/180°. To convert from radians to degrees, multiply by 180°/π.
常见换算：π弧度 = 180°。角度转弧度：乘以π/180°；弧度转角度：乘以180°/π。
2. Exact Values and Signs by Quadrant | 三角函数的精确值与象限符号
IB exams often require exact trigonometric values for special angles. You must memorize the sine, cosine, and tangent of 0°, 30°, 45°, 60°, and 90° (or in radians: 0, π/6, π/4, π/3, π/2).
IB考试经常要求特殊角的三角函数精确值。你必须熟记0°、30°、45°、60°、90°（即弧度0、π/6、π/4、π/3、π/2）的正弦、余弦和正切值。
| Angle | 0° | 30° | 45° | 60° | 90° |
| sin | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan | 0 | √3/3 | 1 | √3 | undefined |
The sign of each trigonometric function depends on the quadrant. A common mnemonic is "All Stations To Central" (ASTC): All positive in QI, Sine positive in QII, Tangent positive in QIII, Cosine positive in QIV.
各三角函数在不同象限的符号不同。常用口诀“ASTC”（All Stations To Central）记忆：第一象限全正，第二象限正弦正，第三象限正切正，第四象限余弦正。
For example, sin 150° = sin 30° = 1/2, and cos 210° = -cos 30° = -√3/2.
例如，sin 150° = sin 30° = 1/2，而cos 210° = -cos 30° = -√3/2。
3. Graphs and Transformations | 三角函数图像与变换
The graphs of y = asin(bx + c) + d and y = acos(bx + c) + d have a vertical stretch/amplitude |a|, a horizontal compression/stretch related to |b|, a phase shift due to -c/b, and a vertical shift d.
y = asin(bx + c) + d 和 y = acos(bx + c) + d 的图像具有垂直伸缩量|a|（振幅）、由|b|决定的水平压缩/拉伸、由-c/b产生的相位平移，以及垂直平移d。
The amplitude is |a|, the period is 2π/|b|, and the principal axis is y = d. For tangent, the period is π/|b|.
振幅为|a|，周期为2π/|b|，平衡轴为y = d。正切函数的周期为π/|b|。
When sketching, identify key points such as maxima, minima, and intercepts. Transformations should be applied in the order: horizontal shift, reflection, stretch, then vertical shift.
画图时，确定最大值、最小值和交点等关键点。变换应按顺序进行：水平平移、反射、伸缩、垂直平移。
4. Basic Identities and Trigonometric Equations | 基本恒等式与三角方程
The three most important Pythagorean identities are:
三个最重要的毕达哥拉斯恒等式是：
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
These identities are used to simplify expressions and solve equations. For example, sin2θ = 1 - cos2θ.
这些恒等式用于化简表达式和解方程。例如，sin²θ = 1 - cos²θ。
When solving a trigonometric equation over a given interval, first find the principal solution, then use the symmetry of the unit circle or the period of the function to find all solutions in the interval.
在给定区间内解三角方程时，先求主解，再利用单位圆的对称性或函数的周期性求出该区间内的所有解。
Example: Solve sin θ = 1/2 for 0 ≤ θ < 2π. The solutions are θ = π/6 and θ = 5π/6.
例：在0 ≤ θ < 2π内解sin θ = 1/2。解为θ = π/6和θ = 5π/6。
5. Compound Angle Formulae | 复合角公式
The compound angle formulae express trigonometric functions of sums and differences of angles:
复合角公式表示角度和与差的三角函数：
sin(A + B) = sin A cos B + cos A sin B
sin(A − B) = sin A cos B − cos A sin B
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
These formulae allow you to find exact values of angles such as 15°, 75°, and 105°. For example, sin 75° = sin(45° + 30°) = sin 45°cos 30° + cos 45°sin 30° = (√6 + √2)/4.
这些公式可用于求15°、75°、105°等角的精确值。例如，sin 75° = sin(45° + 30°) = sin 45°cos 30° + cos 45°sin 30° = (√6 + √2)/4。
6. Double Angle and Half Angle Formulas | 二倍角与半角公式
Double angle formulas are derived from compound angle formulas and are heavily used in integration and equation solving.
二倍角公式由复合角公式推导而来，在积分和解方程中频繁使用。
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ
tan 2θ = 2 tan θ / (1 − tan²θ)
The double angle form of cosine gives the power-reducing identities:
余弦的二倍角公式可推导出降幂恒等式：
sin²θ = (1 − cos 2θ) / 2
cos²θ = (1 + cos 2θ) / 2
These are essential when integrating sin²θ or cos²θ.
这些公式在处理sin²θ或cos²θ的积分时必不可少。
7. Sum-to-Product and Product-to-Sum Formulas | 和差化积与积化和差公式
Product-to-sum identities express products of sine and cosine as sums, which are easier to integrate.
积化和差公式将正弦和余弦的乘积表示为和的形式，从而更容易积分。
sin A cos B = ½ [sin(A+B) + sin(A−B)]
cos A sin B = ½ [sin(A+B) − sin(A−B)]
cos A cos B = ½ [cos(A+B) + cos(A−B)]
sin A sin B = ½ [cos(A−B) − cos(A+B)]
Sum-to-product formulas convert sums into products, useful in solving equations:
和差化积公式将和差转化为乘积，常用于解方程：
sin X + sin Y = 2 sin[(X+Y)/2] cos[(X−Y)/2]
sin X − sin Y = 2 cos[(X+Y)/2] sin[(X−Y)/2]
cos X + cos Y = 2 cos[(X+Y)/2] cos[(X−Y)/2]
cos X − cos Y = −2 sin[(X+Y)/2] sin[(X−Y)/2]
For example, sin 5x + sin x = 2 sin 3x cos 2x, which can help solve equations like sin 5x + sin x = 0.
例如，sin 5x + sin x = 2 sin 3x cos 2x，这有助于解sin 5x + sin x = 0之类的方程。
8. Solving Triangles: Sine and Cosine Rules | 解三角形：正弦定理与余弦定理
For any triangle with sides a, b, c opposite angles A, B, C respectively:
对于任意三角形，边a, b, c分别对角A, B, C：
a / sin A = b / sin B = c / sin C = 2R
c² = a² + b² − 2ab cos C
The sine rule is used when two angles and one side (AAS) or two sides and a non-included angle (SSA) are known. The cosine rule is used for SAS and SSS cases.
当已知两角一边（AAS）或两边及其中一边的对角（SSA）时使用正弦定理；当已知两边夹角（SAS）或三边（SSS）时使用余弦定理。
The area of a triangle can be found using A = 12absin C. This formula works with any two sides and the included angle.
三角形面积可由A = 12absin C求得。该公式适用于已知两边及其夹角的情况。
Be careful with the ambiguous case (SSA): the sine rule may give two possible values for the angle, so always check whether both triangles satisfy the angle sum property.
注意SSA的“双解”问题：正弦定理可能给出两个可能的角度值，必须检查两个三角形是否都满足内角和为180°。
9. Inverse Trigonometric Functions | 反三角函数
Inverse trigonometric functions give the angle whose sine, cosine, or tangent is a given number. Their domains and ranges are restricted to define one-to-one functions.
反三角函数给出正弦、余弦或正切等于给定数值的角度。为了定义一对一函数，它们的定义域和值域被限制。
y = arcsin x has domain -1 ≤ x ≤ 1 and range -π/2 ≤ y ≤ π/2.
y = arcsin x的定义域为-1 ≤ x ≤ 1，值域为-π/2 ≤ y ≤ π/2。
y = arccos x has domain -1 ≤ x ≤ 1 and range 0 ≤ y ≤ π.
y = arccos x的定义域为-1 ≤ x ≤ 1，值域为0 ≤ y ≤ π。
y = arctan x has domain all real numbers and range -π/2 < y < π/2.
y = arctan x的定义域为全体实数，值域为-π/2 < y < π/2。
When solving equations with inverse trig functions, always convert to the principal value unless the question specifies a different interval.
在解含反三角函数的方程时，除非题目指定不同区间，否则应使用主值。
10. Trigonometry in Calculus | 三角函数在微积分中的应用
In IB Analysis and Approaches HL, differentiation and integration of trigonometric functions are core skills.
在IB分析与方法HL中，三角函数的求导和积分是核心技能。
d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (tan x) = sec² x
Using the chain rule, for example, ddxsin(ax+b) = acos(ax+b).
使用链式法则，例如ddxsin(ax+b) = acos(ax+b)。
Integrals of trigonometric functions include ∫ sin xdx = -cos x + C and ∫ cos xdx = sin x + C. For ∫ sin2 xdx, use the identity sin2 x = (1-cos 2x)/2.
三角函数的积分包括∫ sin xdx = -cos x + C和∫ cos xdx = sin x + C。对于∫ sin² xdx，使用恒等式sin² x = (1-cos 2x)/2。
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