📚 Edexcel FP3 Further Pure Mathematics 3: Hyperbolics, Matrices and Vector Geometry | Edexcel FP3 高等纯数3：双曲函数、矩阵与向量几何
Edexcel Further Pure Mathematics 3 (FP3) extends core A-level ideas into hyperbolic functions, advanced coordinate geometry, vector products and matrix diagonalisation. It is one of the more abstract modules, but exam questions are highly structured and reward precise algebraic work.
Edexcel 高等纯数3（FP3）将 A-level 核心知识拓展到双曲函数、高等坐标几何、向量积和矩阵对角化。它是较抽象的模块之一，但考试题型结构清晰，精确的代数运算能稳定得分。
1. Hyperbolic Functions and Their Inverses | 双曲函数及其反函数
Hyperbolic functions are built from exponentials. The three principal functions are sinh, cosh and tanh, and they appear frequently in integration, coordinate geometry and differential equations.
双曲函数由指数函数构造而来。三个主要函数是 sinh、cosh 和 tanh，它们在积分、坐标几何和微分方程中出现频率很高。
sinh x = (eˣ − e⁻ˣ) / 2, cosh x = (eˣ + e⁻ˣ) / 2, tanh x = sinh x / cosh x
Their inverse functions are logarithmic and have strict domains. This is a common source of sign and domain errors in exams.
它们的反函数是对数形式，并且有严格的定义域。这是考试中常见的符号和定义域错误来源。
arsinh x = ln(x + √x² + 1) for all real x
arcosh x = ln(x + √x² − 1) for x ≥ 1
artanh x = ½ ln((1 + x) / (1 − x)) for |x| < 1
2. Hyperbolic Identities and Equations | 双曲恒等式与方程
osh² x − sinh² x = 1. Related forms are 1 − tanh² x = sech² x and coth² x − 1 = cosech² x.
双曲函数的基本恒等式是 cosh² x − sinh² x = 1。相关的形式还有 1 − tanh² x = sech² x 和 coth² x − 1 = cosech² x。
Osborne' ng a product of two sines. For example, cos² θ + sin² θ = 1 becomes cosh² x − sinh² x = 1.
Osborne 规则是一个实用技巧：将 sin 替换为 sinh、cos 替换为 cosh，但任何含有两个正弦乘积的项都要变号。例如 cos² θ + sin² θ = 1 变为 cosh² x − sinh² x = 1。
To solve hyperbolic equations, write everything in terms of eˣ or reduce the equation to a quadratic in eˣ. For example:
解双曲方程时，可以把所有项写成 eˣ 的形式，或将方程化为关于 eˣ 的二次方程。例如：
3 sinh x − 2 cosh x = 1 ⇒ e²ˣ − 2eˣ − 5 = 0
This quadratic factorises or can be solved with the formula, and only the positive root for eˣ is valid.
这个二次方程可以因式分解或用公式求解，并且只有 eˣ 的正根有效。
3. Further Coordinate Systems: Parabola, Ellipse, Hyperbola | 高等坐标系统：抛物线、椭圆、双曲线
FP3 requires confident use of the standard Cartesian and parametric forms of the parabola, ellipse, hyperbola and rectangular hyperbola.
FP3 要求熟练使用抛物线、椭圆、双曲线和等轴双曲线的标准笛卡尔形式和参数形式。
Parabola: y² = 4ax
Ellipse: x²/a² + y²/b² = 1
Hyperbola: x²/a² − y²/b² = 1
Rectangular hyperbola: xy = c²
The standard parametric forms are also exam favourites because tangents and normals are usually easier to derive from them.
标准参数形式也是考试重点，因为从参数形式推导切线和法线通常更容易。
Parabola: (at², 2at)
Ellipse: (a cos θ, b sin θ)
Hyperbola: (a sec θ, b tan θ) or (a cosh t, b sinh t)
Rectangular hyperbola: (ct, c/t)
4. Tangents, Normals and Geometric Properties | 切线、法线与几何性质
For each conic, you should be able to quote or quickly derive the tangent equation at a parameter point. These formulas save time and reduce calculation errors.
对于每种圆锥曲线，你都应该能直接引用或快速推导参数点处的切线方程。这些公式能节省时间并减少计算错误。
Parabola tangent: ty = x + at²
Parabola normal: tx + y = 2at + at³
Ellipse tangent: (x cos θ)/a + (y sin θ)/b = 1
Hyperbola tangent: (x sec θ)/a − (y tan θ)/b = 1
Rectangular hyperbola tangent: x + t² y = 2ct
the ellipse, the foci are (±ae, 0) with e² = 1 − b²/a²; for the hyperbola, e² = 1 + b²/a².
焦点-准线性质也是考点。抛物线 y² = 4ax 的焦点为 (a, 0)，准线为 x = −a。椭圆的焦点为 (±ae, 0)，其中 e² = 1 − b²/a²；双曲线中 e² = 1 + b²/a²。
5. Differentiation of Inverse Functions | 反函数微分
FP3 extends differentiation to inverse trigonometric and inverse hyperbolic functions. These standard results must be learned exactly, including any sign or domain restrictions.
FP3 将微分推广到反三角函数和反双曲函数。这些标准结果必须准确记忆，包括符号和定义域限制。
d/dx arcsin x = 1 / √1 − x²
d/dx arccos x = −1 / √1 − x²
d/dx arctan x = 1 / (1 + x²)
d/dx arsinh x = 1 / √x² + 1
d/dx arcosh x = 1 / √x² − 1
d/dx artanh x = 1 / (1 − x²)
the argument is a function of x, apply the chain rule. For example, d/dx arsinh(2x) = 2 / √4x² + 1.
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