由ƒ(x)= sin(x)定义的ƒ：R⇒[-1，1]范围内的实函数不是双射，因为不同的图像具有相同的图像，例如ƒ(0)= 0，ƒ(2π )= 0，ƒ(π)= 0，因此，ƒ不是一一。由于ƒ不是双射(因为它不是一一)，因此不存在逆。为了使函数双射，我们可以在将函数域限制后将函数的域限制为[-π/ 2，π/ 2]或[-π/ 2，3π/ 2]或[−3π / 2，5π/ 2] ƒ(x)= sin(x)是双射，因此ƒ是可逆的。也就是说，为了使sin(x)可以将其限制为[-π/ 2，π/ 2]或[-π/ 2，3π/ 2]或[-3π/ 2，5π/ 2]或……的域。但是[−π / 2，π/ 2]是sinθ的主要解，因此使sinθ可逆。自然地，如果未提及其他域，则应考虑域[-π/ 2，π/ 2]。
- ƒ：[−π / 2，π/ 2]⇒[-1，1]被定义为ƒ(x)= sin(x)且是双射，因此存在逆。 sin -1的逆也称为反正弦，反函数也称为反弧函数。
- ƒ：[-π/ 2，π/ 2]⇒[-1，1]定义为sinθ= x⇔sin -1 (x)=θ，θ属于[-π/ 2，π/ 2]和x属于[−1，1]。
同样，我们限制cos，tan，cot，sec，cosec的域，以便它们是可逆的。以下是一些三角函数及其域和范围。
|
Function |
Domain |
Range |
|---|---|---|
| sin-1 | [ -1 , 1 ] | [ −π/2 , π/2 ] |
| cos-1 | [ -1 , 1 ] | [ 0 , π ] |
| tan-1 | R | [ −π/2 , π/2 ] |
| cot-1 | R | [ 0 , π ] |
| sec-1 | ( -∞ , -1 ] U [ 1,∞ ) | [ 0 , π ] − { π/2 } |
| cosec-1 | ( -∞ , -1 ] U [ 1 , ∞ ) | [ −π/2 , π/2 ] – {0} |
反三角函数的性质
第一组：罪的性质
1) sin(θ) = x ⇔ sin-1(x) = θ , θ ∈ [ -π/2 , π/2 ], x ∈ [ -1 , 1 ]
2) sin-1(sin(θ)) = θ , θ ∈ [ -π/2 , π/2 ]
3) sin(sin-1(x)) = x , x ∈ [ -1 , 1 ]
例子：
- sin(π/6) = 1/2 ⇒ sin-1(1/2) = π/6
- sin-1(sin(π/6)) = π/6
- sin(sin-1(1/2)) = 1/2
第2集：cos的性质
4) cos(θ) = x ⇔ cos-1(x) = θ , θ ∈ [ 0 , π ] , x ∈ [ -1 , 1 ]
5) cos-1(cos(θ)) = θ , θ ∈ [ 0 , π ]
6) cos(cos-1(x)) = x , x ∈ [ -1 , 1 ]
例子：
- cos(π/6) = √3/2 ⇒ cos-1(√3/2) = π/6
- cos-1(cos(π/6)) = π/6
- cos(cos-1(1/2)) = 1/2
第3组：棕褐色的属性
7) tan(θ) = x ⇔ tan-1(x) = θ , θ ∈ [ -π/2 , π/2 ] , x ∈ R
8) tan-1(tan(θ)) = θ , θ ∈ [ -π/2 , π/2 ]
9) tan(tan-1(x)) = x , x ∈ R
例子：
- tan(π/4) = 1 ⇒ tan-1(1) = π/4
- tan-1(tan(π/4)) = π/4
- tan(tan-1(1)) = 1
第4组：婴儿床的属性
10) cot(θ) = x ⇔ cot-1(x) = θ , θ ∈ [ 0 , π ] , x ∈ R
11) cot-1(cot(θ)) = θ , θ ∈ [ 0 , π ]
12) cot(cot-1(x)) = x , x ∈ R
例子：
- cot(π/4) = 1 ⇒ cot-1(1) = π/4
- cot(cot-1(π/4)) = π/4
- cot(cot(1)) = 1
设置5：秒的属性
13) sec(θ) = x ⇔ sec-1(x) = θ , θ ∈ [ 0 , π] – { π/2 } , x ∈ (-∞,-1] ∪ [1,∞)
14) sec-1(sec(θ)) = θ , θ ∈ [ 0 , π] – { π/2 }
15) sec(sec-1(x)) = x , x ∈ ( -∞ , -1 ] ∪ [ 1 , ∞ )
例子：
- sec(π/3) = 1/2 ⇒ sec-1(1/2) = π/3
- sec-1(sec(π/3)) = π/3
- sec(sec-1(1/2)) = 1/2
设置6：cosec的属性
16) cosec(θ) = x ⇔ cosec-1(x) = θ , θ ∈ [ -π/2 , π/2 ] – { 0 } , x ∈ ( -∞ , -1 ] ∪ [ 1,∞ )
17) cosec-1(cosec(θ)) = θ , θ ∈[ -π/2 , π ] – { 0 }
18) cosec(cosec-1(x)) = x , x ∈ ( -∞,-1 ] ∪ [ 1,∞ )
例子：
- cosec(π/6) = 2 ⇒ cosec-1(2) = π/6
- cosec-1(cosec(π/6)) = π/6
- cosec(cosec-1(2)) = 2
集合7：其他反三角公式
19) sin-1(-x) = -sin-1(x) , x ∈ [ -1 , 1 ]
20) cos-1(-x) = π – cos-1(x) , x ∈ [ -1 , 1 ]
21) tan-1(-x) = -tan-1(x) , x ∈ R
22) cot-1(-x) = π – cot-1(x) , x ∈ R
23) sec-1(-x) = π – sec-1(x) , x ∈ ( -∞ , -1 ] ∪ [ 1 , ∞ )
24) cosec-1(-x) = -cosec-1(x) , x ∈ ( -∞ , -1 ] ∪ [ 1 , ∞ )
例子：
- sin-1(-1/2) = -sin-1(1/2)
- cos-1(-1/2) = π -cos-1(1/2)
- tan-1(-1) = π -tan(1)
- cot-1(-1) = -cot-1(1)
- sec-1(-2) = -sec-1
集合8：两个三角函数的和
25) sin-1(x) + cos-1(x) = π/2 , x ∈ [ -1 , 1 ]
26) tan-1(x) + cot-1(x) = π/2 , x ∈ R
27) sec-1(x) + cosec-1(x) = π/2 , x ∈ ( -∞ , -1 ] ∪ [ 1 , ∞ )
证明：
sin-1(x) + cos-1(x) = π/2 , x ∈ [ -1 , 1 ]
let sin-1(x) = y
now,
x = sin y = cos((π/2) − y)
⇒ cos-1(x) = (π/2) – y = (π/2) −sin-1(x)
so, sin-1(x) + cos-1(x) = π/2
tan-1(x) + cot-1(x) = π/2 , x ∈ R
Let tan-1(x) = y
now, cot(π/2 − y) = x
⇒ cot-1(x) = (π/2 − y)
tan-1(x) + cot-1(x) = y + π/2 − y
so, tan-1(x) + cot-1(x) = π/2
同样，我们也可以证明arcsec和arccosec之和的定理。
设置9：三角函数的转换
28) sin-1(1/x) = cosec-1(x) , x≥1 or x≤−1
29) cos-1(1/x) = sec-1(x) , x ≥ 1 or x ≤ −1
30) tan-1(1/x) = −π + cot-1(x)
证明：
sin-1(1/x) = cosec-1(x) , x ≥ 1 or x ≤ −1
let, x = cosec(y)
1/x = sin(y)
⇒ sin-1(1/x) = y
⇒ sin-1(1/x) = cosec-1(x)
同样，我们也可以证明arccos和arctan定理
例子：
sin-1(1/2) = cosec-1(2)
设置10：定期函数转换
arcsin(x) = (-1)n arcsin(x) + πn
arccos(x) = ±arc cos x + 2πn
arctan(x) = arctan(x) + πn
arccot(x) = arccot(x) + πn
where n = 0, ±1, ±2, ….