A real function in the range ƒ : R ⇒ [-1 , 1] defined by ƒ(x) = sin(x) is not a bijection since different images have the same image such as ƒ(0) = 0, ƒ(2π) = 0,ƒ(π) = 0, so, ƒ is not one-one. Since ƒ is not a bijection (because it is not one-one) therefore inverse does not exist. To make a function bijective we can restrict the domain of the function to [−π/2, π/2] or [−π/2, 3π/2] or [−3π/2, 5π/2] after restriction of domain ƒ(x) = sin(x) is a bijection, therefore ƒ is invertible. i.e. to make sin(x) we can restrict it to the domain [−π/2, π/2] or [−π/2, 3π/2] or [−3π/2, 5π/2] or……. but [−π/2, π/2] is the Principal solution of sinθ, hence to make sinθ invertible. Naturally, the domain [−π/2, π/2] should be considered if no other domain is mentioned.
- ƒ: [−π/2, π/2] ⇒ [-1, 1] is defined as ƒ(x) = sin(x) and is a bijection, hence inverse exists. The inverse of sin-1 is also called arcsine and inverse functions are also called arc functions.
- ƒ:[−π/2 , π/2] ⇒ [−1 , 1] is defined as sinθ = x ⇔ sin-1(x) = θ , θ belongs to [−π/2 , π/2] and x belongs to [−1 , 1] .
Similarly, we restrict the domains of cos, tan, cot, sec, cosec so that they are invertible. Below are some trigonometric functions with their domain and range.
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} |
Properties of Inverse Trigonometric Functions
Set 1: Properties of sin
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 ]
Examples:
- 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
Set 2: Properties of cos
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 ]
Examples:
- 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
Set 3: Properties of tan
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
Examples:
- tan(π/4) = 1 ⇒ tan-1(1) = π/4
- tan-1(tan(π/4)) = π/4
- tan(tan-1(1)) = 1
Set 4: Properties of cot
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
Examples:
- cot(π/4) = 1 ⇒ cot-1(1) = π/4
- cot(cot-1(π/4)) = π/4
- cot(cot(1)) = 1
Set 5: Properties of sec
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 , ∞ )
Examples:
- 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
Set 6: Properties of cosec
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,∞ )
Examples:
- cosec(π/6) = 2 ⇒ cosec-1(2) = π/6
- cosec-1(cosec(π/6)) = π/6
- cosec(cosec-1(2)) = 2
Set 7: Other inverse trigonometric formulas
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 , ∞ )
Examples:
- 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
Set 8: Sum of two trigonometric functions
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 , ∞ )
Proof:
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
Similarly, we can prove the theorem of the sum of arcsec and arccosec as well.
Set 9: Conversion of trigonometric functions
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)
Proof:
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)
Similarly, we can prove the theorem of arccos and arctan as well
Example:
sin-1(1/2) = cosec-1(2)
Set 10: Periodic functions conversion
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, ….