反三角函数定义域图像求导全解析
反三角函数是Edexcel A-Level数学Pure 2部分的重要考点：arcsin x、arccos x、arctan x三兄弟的定义域与值域、图像特征、求导公式和标准积分，几乎每年都会出现在真题中。本文从"为什么sin没有反函数"讲起，逐一推导三个反三角函数的导数公式，总结Edexcel真题的典型题型与高频易错点，帮助考生在考试中稳定拿分。
Inverse trigonometric functions are a key topic in the Pure 2 component of Edexcel A-Level Mathematics. The domains and ranges, graph shapes, derivative formulas and standard integrals of arcsin x, arccos x and arctan x appear in the exam almost every year. This article starts from the question of why sin x has no inverse, derives the derivative of each inverse trigonometric function step by step, and summarises the typical question types and high-frequency mistakes in Edexcel papers, so that students can secure full marks in the exam.
一、什么是反三角函数：为什么sin x没有反函数 | Why sin x Has No Inverse Function
反三角函数解决的问题是"已知三角函数值，求对应的角"。比如sin x = 1/2，x等于多少？初中我们就知道x可以是30度，但sin 150度也等于1/2，sin 390度还是1/2。这是因为sin x是周期函数，同一个y值对应无穷多个x值，函数图像上一条水平线会与图像相交无数次，不满足"一个输入对应一个输出"的一一对应条件，所以整个sin x没有反函数。Edexcel教材的解决办法是限制定义域：把sin x限制在[-π/2, π/2]这个区间上，此时函数单调递增，每个y值恰好对应一个x值，反函数就存在了，记作arcsin x，也可以写成sin⁻¹x。类似地，cos x限制在[0, π]，tan x限制在(-π/2, π/2)，分别得到arccos x和arctan x。
Inverse trigonometric functions answer the question: given a trigonometric value, what is the corresponding angle? For example, if sin x = 1/2, what is x? At GCSE level we know x could be 30 degrees, but sin 150 degrees is also 1/2, and so is sin 390 degrees. Because sin x is a periodic function, one value of y corresponds to infinitely many values of x, and a horizontal line crosses the graph infinitely many times. The function is not one-to-one, so sin x has no inverse on its whole domain. The Edexcel solution is to restrict the domain: on the interval [-pi/2, pi/2] the function is strictly increasing, each y value matches exactly one x value, and an inverse exists, written arcsin x or sin to the power minus one. Similarly, cos x is restricted to [0, pi] to give arccos x, and tan x is restricted to (-pi/2, pi/2) to give arctan x.
二、arcsin x的定义域与值域：为什么答案必须落在[-π/2, π/2] | Domain and Range of arcsin x
反函数的定义域等于原函数的值域，反函数的值域等于原函数的定义域。sin x限制在[-π/2, π/2]后值域是[-1, 1]，所以arcsin x的定义域是[-1, 1]，值域是[-π/2, π/2]。这个结论必须背熟，因为Edexcel真题常常直接考查：arcsin(1/2)等于多少？正确答案是π/6，因为π/6落在值域[-π/2, π/2]内；而5π/6虽然也满足sin(5π/6) = 1/2，但不在值域内，不能作为答案。再记三个特殊值：arcsin 0 = 0，arcsin 1 = π/2，arcsin(-1) = -π/2。注意arcsin x的输入必须是[-1, 1]之间的数，如果题目中出现arcsin 2或者arcsin(-1.5)，答案直接是"无定义"。
The domain of an inverse function equals the range of the original function, and its range equals the domain of the original function. Since sin x restricted to [-pi/2, pi/2] has range [-1, 1], arcsin x has domain [-1, 1] and range [-pi/2, pi/2]. This fact must be memorised because Edexcel questions test it directly: what is arcsin(1/2)? The correct answer is pi/6, because pi/6 lies inside the range [-pi/2, pi/2], while 5pi/6 also satisfies sin(5pi/6) = 1/2 but is outside the range and therefore cannot be the answer. Three special values are worth remembering: arcsin 0 = 0, arcsin 1 = pi/2 and arcsin(-1) = -pi/2. Note that the input of arcsin x must lie between -1 and 1; if a question contains arcsin 2 or arcsin(-1.5), the answer is simply undefined.
三、arccos x与arctan x的图像特征：递减曲线与水平渐近线 | Graphs of arccos x and arctan x
三个反三角函数图像各有一个突出的视觉特征。arcsin x的图像是上升的S形，经过原点，定义域[-1, 1]，两端点分别是(-1, -π/2)和(1, π/2)。arccos x的图像则是下降的曲线，定义域同样是[-1, 1]，但值域是[0, π]，两个端点是(-1, π)和(1, 0)，图像经过(0, π/2)。arctan x最特别：定义域是全体实数，图像在x趋于正无穷时无限靠近水平渐近线y = π/2，在x趋于负无穷时无限靠近y = -π/2，但永远不触及；图像关于原点对称，是奇函数，满足arctan(-x) = -arctan(x)，且arctan x在x = 0处过原点。画图时记住三个关键点：arcsin过原点且递增，arccos递减，arctan有两条渐近线。
Each of the three inverse trigonometric graphs has one distinctive visual feature. The graph of arcsin x is a rising S-shape passing through the origin, with domain [-1, 1] and endpoints (-1, -pi/2) and (1, pi/2). The graph of arccos x is a falling curve with the same domain [-1, 1] but range [0, pi]; its endpoints are (-1, pi) and (1, 0), and it passes through (0, pi/2). arctan x is the most unusual: its domain is all real numbers, the graph approaches the horizontal asymptote y = pi/2 as x tends to positive infinity and approaches y = -pi/2 as x tends to negative infinity, without ever touching either line. The graph is symmetric about the origin, being an odd function with arctan(-x) = -arctan(x), and it passes through the origin. When sketching, remember three key points: arcsin rises through the origin, arccos falls, and arctan has two horizontal asymptotes.
四、arcsin x的导数公式推导：隐函数求导的两步法 | Differentiating arcsin x by Implicit Differentiation
反三角函数的导数不能用四则运算法则直接求，Edexcel Pure 2使用隐函数求导法。设y = arcsin x，两边同时取sin得sin y = x。现在把y看成x的函数，对等式两边关于x求导：左边用链式法则，cos y乘以dy/dx；右边x的导数是1，得到cos y · dy/dx = 1，所以dy/dx = 1/cos y。最后把cos y换成x的表达式：由sin y = x和恒等式sin²y + cos²y = 1，得cos y = √(1 - x²)，注意因为y在[-π/2, π/2]内，cos y恒为非负，根号前取正号。于是arcsin x的导数等于1/√(1 - x²)，这个公式必须记牢。注意导数只在开区间(-1, 1)内有定义，在x = ±1处分母为零，导数趋于无穷大，图像在这两端是竖直切线。
The derivatives of inverse trigonometric functions cannot be found by the usual rules of differentiation, so Edexcel Pure 2 uses implicit differentiation. Let y = arcsin x and take sine of both sides to get sin y = x. Treating y as a function of x and differentiating both sides with respect to x, the left side requires the chain rule, giving cos y times dy/dx, while the right side differentiates to 1. Hence cos y multiplied by dy/dx equals 1, so dy/dx = 1/cos y. Finally, express cos y in terms of x: from sin y = x and the identity sin squared y plus cos squared y = 1, we obtain cos y = the square root of (1 - x squared), taking the positive root because y lies in [-pi/2, pi/2], where cos y is never negative. Therefore the derivative of arcsin x is 1 over the square root of (1 - x squared), a formula worth memorising. Note that the derivative is defined only on the open interval (-1, 1); at x = plus or minus 1 the denominator is zero, the derivative becomes infinite, and the graph has vertical tangents at both ends.
五、arccos x与arctan x的导数：负号与根号的两个记忆点 | Derivatives of arccos x and arctan x
用同样的隐函数求导法可以得到另外两个导数。设y = arccos x，则cos y = x，两边求导得-sin y · dy/dx = 1，于是dy/dx = -1/sin y = -1/√(1 - x²)，即arccos x的导数是负的1/√(1 - x²)。设y = arctan x，则tan y = x，两边求导得sec²y · dy/dx = 1，利用sec²y = 1 + tan²y = 1 + x²，得到dy/dx = 1/(1 + x²)。两个记忆技巧：第一，arcsin x和arccos x的导数分母完全相同，只有正负号之差，arcsin取正号，arccos取负号，符号错误是Edexcel考生最常丢分的地方；第二，arctan x的导数分母没有根号，是1 + x²，而且对任意实数x都有定义，不像前两个在x = ±1处出问题。三组公式放在一起对比记忆，考试时直接调用。
The same implicit differentiation method produces the other two derivatives. Let y = arccos x, so cos y = x; differentiating gives minus sin y times dy/dx = 1, hence dy/dx = -1/sin y = -1 over the square root of (1 - x squared). Let y = arctan x, so tan y = x; differentiating gives sec squared y times dy/dx = 1, and using sec squared y = 1 + tan squared y = 1 + x squared yields dy/dx = 1/(1 + x squared). Two memory aids help: first, arcsin x and arccos x have identical denominators in their derivatives and differ only by sign, with arcsin positive and arccos negative, and sign errors are among the most common marks lost by Edexcel candidates; second, the derivative of arctan x has no square root in the denominator, being simply 1 + x squared, and it is defined for every real x, unlike the other two which fail at x = plus or minus 1. Comparing the three formulas side by side makes them much easier to recall in the exam.
六、标准积分公式：∫1/√(1-x²)dx与∫1/(1+x²)dx | Standard Integrals Involving Inverse Trigonometric Functions
导数公式反过来就是积分公式，Edexcel公式书把这三组标准积分列为必背内容：∫1/√(1 - x²) dx = arcsin x + C，∫1/(1 + x²) dx = arctan x + C，而∫-1/√(1 - x²) dx = arccos x + C。实际解题中arcsin和arctan两个公式用得最多。考试还常考带系数a的变体：∫1/√(a² - x²) dx = arcsin(x/a) + C，∫1/(a² + x²) dx = (1/a)·arctan(x/a) + C，这两个变体本质上是换元u = x/a的结果，建议直接记住。判断用哪个公式有一个快速方法：分母是√(1 - x²)或√(a² - x²)形式，结果是arcsin；分母是1 + x²或a² + x²形式，结果是arctan。遇到这种形式的积分，先看分母结构再套公式，不要试图用三角换元从头推导，考试时间不等人。
Reversing the derivative formulas gives the standard integrals, and the Edexcel formula booklet lists them as essential: the integral of 1 over the square root of (1 - x squared) dx equals arcsin x plus C, the integral of 1 over (1 + x squared) dx equals arctan x plus C, and the integral of -1 over the square root of (1 - x squared) dx equals arccos x plus C. In practice the arcsin and arctan formulas are used most often. Examiners also like the scaled variants: the integral of 1 over the square root of (a squared - x squared) dx equals arcsin(x/a) plus C, and the integral of 1 over (a squared + x squared) dx equals (1/a) times arctan(x/a) plus C. Both follow from the substitution u = x/a and are worth memorising directly. A quick way to choose the right formula: if the denominator is the square root of (1 - x squared) or the square root of (a squared - x squared), the result is arcsin; if the denominator is 1 + x squared or a squared + x squared, the result is arctan. Look at the denominator structure first and apply the formula directly, rather than re-deriving from scratch with trigonometric substitution.
七、Edexcel真题典型题型：求值、复合求导与定积分 | Typical Edexcel Question Types
反三角函数在Edexcel真题中主要有三类考法。第一类是求值题：直接计算arcsin(√3/2)、arctan 1这类特殊值，考查定义域与值域的理解，答案是π/3和π/4，一定要用弧度制作答。第二类是复合函数求导：例如求y = arcsin(2x)的导数，用链式法则，外层导数1/√(1 - (2x)²)，再乘以内层导数2，结果是2/√(1 - 4x²)。再如y = arctan(x²)，结果是2x/(1 + x⁴)。这类题的关键是先看清内层函数，再套公式。第三类是定积分：例如∫从0到1的1/(1 + x²)dx，直接套arctan公式，结果等于arctan 1 - arctan 0 = π/4 - 0 = π/4，这是近十年真题反复出现的答案。遇到积分上限或下限是特殊值时，要能熟练写出对应角度。
Inverse trigonometric functions appear in Edexcel papers in three main ways. The first is direct evaluation: computing special values such as arcsin(root 3 over 2) or arctan 1, which test understanding of domain and range; the answers are pi/3 and pi/4, always given in radians. The second is differentiating composite functions: for example, to differentiate y = arcsin(2x), apply the chain rule, taking the outer derivative 1 over the square root of (1 - (2x) squared) and multiplying by the inner derivative 2, giving 2 over the square root of (1 - 4x squared). Similarly, the derivative of y = arctan(x squared) is 2x/(1 + x to the fourth). The key is to identify the inner function before applying the formula. The third is definite integration: for example, the integral from 0 to 1 of 1/(1 + x squared) dx uses the arctan formula directly and equals arctan 1 minus arctan 0, which is pi/4 minus 0, giving pi/4, an answer that has appeared repeatedly in papers over the past decade. When the limits are special values, students should write the corresponding angles fluently.
八、高频易错点：定义域陷阱与计算器模式 | Common Mistakes: Domain Traps and Calculator Mode
反三角函数题失分往往不是因为公式不会，而是细节出错。第一个陷阱是定义域：arcsin x的输入必须在[-1, 1]内，看到arcsin 2直接判断无定义；而arctan x对任何实数都有意义，不要混淆。第二个陷阱是符号混淆：arcsin x是"反三角函数"，完全不同于1/sin x（即cosec x），两者写法相近但含义天差地别，Edexcel考生经常在这里丢分。第三个陷阱是复合函数：arcsin(sin x)不一定等于x，例如x = 3π/4时，sin(3π/4) = √2/2，arcsin(√2/2) = π/4，结果落在值域[-π/2, π/2]内，而不是原来的3π/4。第四个陷阱是计算器模式：求反三角函数时必须把计算器调到弧度模式，用角度模式算出的数值全部错误；同理，求定积分答案写成角度制也是零分。
Marks are usually lost on inverse trigonometric questions through small details rather than missing formulas. The first trap is the domain: the input of arcsin x must lie between -1 and 1, so arcsin 2 is simply undefined, while arctan x accepts every real number, so do not confuse the two. The second trap is notation: arcsin x is an inverse trigonometric function and is completely different from 1 over sin x, which is cosec x; the two look similar but mean entirely different things, and Edexcel candidates frequently lose marks here. The third trap is composition: arcsin(sin x) does not always equal x. For example, when x = 3pi/4, sin(3pi/4) = root 2 over 2, and arcsin of that value is pi/4, which lies in the range [-pi/2, pi/2], not the original 3pi/4. The fourth trap is the calculator mode: when evaluating inverse trigonometric functions the calculator must be set to radians; degree mode gives completely wrong values, and writing a definite integral answer in degrees earns zero marks.
九、复习建议与考试技巧：公式对比表与真题训练 | Revision Advice and Exam Technique
反三角函数内容不多，但公式密集，建议按三步复习。第一步，把三组知识做成一张对比表：arcsin x的定义域[-1, 1]、值域[-π/2, π/2]；arccos x的定义域[-1, 1]、值域[0, π]；arctan x的定义域全体实数、值域(-π/2, π/2)，以及对应的导数公式，贴在书桌上每天过一遍。第二步，动手画三张图像，标注关键点、渐近线和单调性，图像记忆比死记文字更牢固。第三步，用Edexcel近五年的past paper真题做专项训练，把求值、复合求导、定积分三类题型各练熟10道左右，特别关注每年Paper 2的第8到第12题附近的反三角函数小题。考试时先看题目考查的是求值、求导还是积分，再决定调用哪组公式；遇到带系数的积分，优先套a²变体公式，节省时间。
Inverse trigonometric functions cover a small amount of content but a dense set of formulas, so revise in three steps. First, build a comparison table of the three groups: arcsin x with domain [-1, 1] and range [-pi/2, pi/2], arccos x with domain [-1, 1] and range [0, pi], and arctan x with domain all real numbers and range (-pi/2, pi/2), together with their derivative formulas, and review it daily. Second, sketch the three graphs by hand, marking key points, asymptotes and monotonicity; visual memory is far more reliable than memorising text. Third, practise with Edexcel past papers from the last five years, working through about ten questions of each type: evaluation, composite differentiation and definite integration, paying special attention to the small inverse trigonometric questions near questions 8 to 12 of Paper 2 each year. In the exam, first identify whether the question asks for evaluation, differentiation or integration, then choose the matching formula set; for integrals with a coefficient, apply the scaled a squared variant directly to save time.
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