📚 Further Trigonometric Functions | 更多三角函数
Building on the basic trigonometric functions, IB Mathematics introduces further trigonometric functions including inverse trigonometric functions (arcsin, arccos, arctan) and reciprocal functions (sec, csc, cot). These functions are essential for solving equations, analysing periodic behaviour, and extending the toolkit of calculus.
在基本三角函数的基础上，IB数学引入了更多三角函数，包括反三角函数 (arcsin, arccos, arctan) 和倒数三角函数 (sec, csc, cot)。这些函数对于求解方程、分析周期行为以及扩展微积分工具箱至关重要。
1. Introduction to Further Trigonometric Functions | 更多三角函数简介
c x = 1/cos x, csc x = 1/sin x and cot x = cos x/sin x play important roles in identities and calculus.
标准的三角函数 sin x, cos x 和 tan x 是周期函数，在整个定义域内不是一一映射的，因此它们的反函数无法全局定义。通过限制定义域，我们得到了反函数 arcsin x, arccos x 和 arctan x。此外，倒数函数 sec x = 1/cos x, csc x = 1/sin x 以及 cot x = cos x/sin x 在恒等式和微积分中扮演着重要的角色。
2. The Inverse Sine Function (arcsin) | 反正弦函数
y first restricting the domain of sin x to [-π/2, π/2]. On this interval, sine is strictly increasing and takes all values from -1 to 1, so the inverse exists.
反正弦函数记为 y = arcsin x 或 y = sin⁻¹ x，定义时首先将 sin x 的定义域限制在 [-π/2, π/2] 上。在这个区间上，正弦严格单调递增，且取遍从 -1 到 1 的所有值，因此反函数存在。
nge [-π/2, π/2], passing through the origin with an increasing shape and two horizontal ends.
y = arcsin x 的图像是 y = sin x（限制域）关于直线 y = x 的对称图形。它的定义域是 [-1, 1]，值域是 [-π/2, π/2]，图像经过原点，整体呈上升形态且两端趋于水平。
3. The Inverse Cosine Function (arccos) | 反余弦函数
estricted to [0, π]. In this interval, cosine is strictly decreasing, covering all values from 1 down to -1. Therefore the inverse cosine function is well-defined with domain [-1, 1] and range [0, π].
对于 y = arccos x，cos x 的定义域被限制在 [0, π]。在这个区间上，余弦严格单调递减，覆盖了从 1 到 -1 的所有值。因此反余弦函数定义良好，其定义域是 [-1, 1]，值域是 [0, π]。
The graph of arccos x decreases from (1, 0) to (-1, π), showing a smooth decline. Unlike arcsin x, its range does not include negative angles, and it is not an odd function.
y = arccos x 的图像从点 (1, 0) 下降至 (-1, π)，呈现平滑的下降趋势。与 arcsin x 不同，它的值域不包含负角，且不是奇函数。
4. The Inverse Tangent Function (arctan) | 反正切函数
nterval (-π/2, π/2). Since tan x approaches ±∞ near the asymptotes, its inverse has domain ℝ raph of y = arctan x is an S-shaped curve with horizontal asymptotes y = -π/2 and y = π/2.
反正切函数是将 tan x 的定义域限制在开区间 (-π/2, π/2) 而得到的。由于 tan x 在渐近线附近趋向 ±∞，其反函数的定义域为全体实数 ℝ，值域为 (-π/2, π/2)。y = arctan x 的图像是一条 S 形曲线，具有水平渐近线 y = -π/2 和 y = π/2。
Arctan x is an odd function and passes through the origin with a slope of 1. It is commonly used to convert gradients into angles and appears in many integration formulas.
arctan x 是奇函数，经过原点且斜率为 1。它常用于由斜率求角度，并出现在许多积分公式中。
5. Domain and Range of Inverse Trig Functions | 反三角函数的定义域和值域
The following table summarises the domain and range for the three principal inverse trigonometric functions.
下表总结了三个主要反三角函数的定义域和值域。
| Function | Domain | Range |
|---|---|---|
| y = arcsin x | [-1, 1] | [-π/2, π/2] |
| y = arccos x | [-1, 1] | [0, π] |
| y = arctan x | ℝ | (-π/2, π/2) |
Memorising these restrictions is vital when solving equations or composing functions, as the output angle must always lie in the correct principal range.
记住这些限制对于解方程或进行函数复合至关重要，因为输出的角必须始终落在正确的主值区间内。
6. Properties and Identities of Inverse Functions | 反函数的性质与恒等式
For values of x in the appropriate domains, we have direct cancellation identities:
在适当的定义域内，我们有直接的消去恒等式：
- sin(arcsin x) = x for x ∈ [-1, 1]
- sin(arcsin x) = x 对于 x ∈ [-1, 1] 成立
- cos(arccos x) = x for x ∈ [-1, 1]
- cos(arccos x) = x 对于 x ∈ [-1, 1] 成立
- tan(arctan x) = x for all real x
- tan(arctan x) = x 对于所有实数 x 成立
In the opposite direction, arcsin(sin x) = x only when x ∈ [-π/2, π/2], and similarly arccos(cos x) = x for x ∈ [0, π] and arctan(tan x) = x for x ∈ , π/2). Outside these intervals the output is adjusted to fall within the principal range.
反过来，arcsin(sin x) = x 仅在 x ∈ [-π/2, π/2] 时成立；类似地，arccos(cos x) = x 对于 x ∈ [0, π]，而 arctan(tan x) = x 对于 x ∈ (-π/2, π/2)。超出这些区间，输出会被调整到主值范围内。
A crucial complementary identity is arcsin x + arccos x = π/2 for all x ∈ [-1, 1]. This relationship is often used to simplify expressions or to solve equations.
一个重要的互补恒等式是 arcsin x + arccos x = π/2，对所有 x ∈ [-1, 1] 成立。这一关系常用于化简表达式或解方程。
7. The Reciprocal Trigonometric Functions | 倒数三角函数（sec, csc, cot）
The reciprocal trigonometric functions are defined directly from the basic sine, cosine and tangent:
倒数三角函数直接由基本的正弦、余弦和正切定义：
- sec x = 1 / cos x (undefined when cos x = 0)
- sec x = 1 / cos x（当 cos x = 0 时无定义）
- csc x = 1 / sin x (undefined when sin x = 0)
- csc x = 1 / sin x（当 sin x = 0 时无定义）
- cot x = cos x / sin x = 1 / tan x (undefined when sin x = 0)
- cot x = cos x / sin x = 1 / tan x（当 sin x = 0 时无定义）
1 + cot² x = csc² x, which are derived from the Pythagorean identity sin² x + cos² x = 1.
这些函数扩展了三角函数的工具包，并引出了有用的恒等式，例如 1 + tan² x = sec² x 和 1 + cot² x = csc² x，它们可由毕达哥拉斯恒等式 sin² x + cos² x = 1 推导得出。
8. Graphs of sec x, csc x, and cot x | 正割、余割、余切函数的图像
π/2 + kπ. It consists of U-shaped branches with minimum value 1 or maximum value -1, and a period of 2π.
y = sec x 的图像在 cos x = 0 处有垂直渐近线，即 x = π/2 + kπ。图像由一系列 U 形分支组成，最小值为 1 或最大值为 -1，周期为 2π。
= 0. Its branches open upward when sin x > 0 and downward when sin x < 0, with period 2π.
y = csc x 的图像类似，在 sin x = 0 即 x = kπ 处有渐近线。当 sin x > 0 时分支开口向上，当 sin x < 0 时分支开口向下，周期也是 2π。
π/2 + kπ, and is strictly decreasing on each interval between asymptotes. Its period is π.
余切函数 y = cot x 在 x = kπ 处有渐近线，在 x = π/2 + kπ 处穿过 x 轴，并且在渐近线之间的每个区间上严格递减。其周期为 π。
9. Solving Equations with Inverse and Reciprocal Functions | 含反三角函数和倒数三角函数的方程求解
equire careful consideration of domain and range. For example, solve arcsin(2x - 1) = π/6.
涉及反三角函数的方程常常需要仔细考虑定义域和值域。例如，求解方程 arcsin(2x - 1) = π/6。
d x = 3/4. Since x = 3/4 gives 2x - 1 = 1/2, which lies in [-1, 1], the solution is valid.
对方程两边取正弦，得到 2x - 1 = sin(π/6) = 1/2，因此 2x = 3/2，x = 3/4。由于 x = 3/4 给出 2x - 1 = 1/2，该值属于 [-1, 1]，所以解有效。
ions, such as sec x = 2, rewrite as cos x = 1/2. The general solution is x = ±π/3 + 2kπ, k ∈ ℤ. Always check for extraneous solutions caused by domain restrictions.
当解含倒数函数的方程时，例如 sec x = 2，可重写为 cos x = 1/2。通解为 x = ±π/3 + 2kπ, k ∈ ℤ。务必检查由于定义域限制可能产生的增根。
10. Composite Functions and Applications | 复合函数及其应用
= arcsin x, then sin θ = x and we place x as opposite over hypotenuse 1, giving adjacent = √1 - x². Hence tan(arcsin x) = x / √1 - x².
诸如 tan(arcsin x) 这样的复合函数可以通过构造直角三角形来化简。设 θ = arcsin x，则 sin θ = x，将对边设为 x、斜边设为 1，则邻边 = √1 - x²。因此 tan(arcsin x) = x / √1 - x²。
This technique is extremely useful in calculus when integrating expressions involving √a² - x² or similar radicals, and for proving identities. Combined with reciprocal functions, one can handle a much wider class of problems.
这一技巧在微积分中对含有 √a² - x² 或类似根式的积分非常有用，也可用于证明恒等式。结合倒数函数，可以处理更广泛的一类问题。
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