Inverse trigonometric functions, also known as arc functions, are used to find the angle that corresponds to a given trigonometric value. These functions are the inverses of the six main trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. In this article, we will explore the key properties of inverse trigonometric functions, including their domains, ranges, and unique characteristics.
Table of Contents
What are Inverse Trigonometric Functions?
Inverse trigonometric functions are the inverse functions of the basic trigonometric functions, which are sine, cosine, tangent, cotangent, secant, and cosecant.
These functions are used to find the angle of a triangle from any of the trigonometric ratios. They are also known as arcus functions, anti-trigonometric functions, or cyclometric functions
Restriction on Trigonometric Functions
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 sin θ, hence to make sin θ 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.
Domain and Range of Inverse Trigonometric Functions
Below are some inverse 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
There are various properties of inverse trigonometric functions that are discussed as follows:
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, . . .
FAQs on Properties of Inverse Trigonometric Functions
What are inverse trigonometric functions?
Inverse trigonometric functions find angles when given trigonometric values, reversing sine, cosine, tangent, etc.
What are the domains and ranges of inverse trigonometric functions?
- Arcsine (sin⁻¹x): Domain [−1,1], Range [−π/2, π/2]
- Arccosine (cos⁻¹x): Domain [−1,1], Range [0,π]
- Arctangent (tan⁻¹x): Domain (−∞,∞), Range (−π/2, π/2)
What are the principal values of inverse trigonometric functions?
Principal values are the main output values within specified ranges for arcsine, arccosine, and arctangent.
What are the properties of arcsine (sin-1 x)?
Some of the properties of inverse of sin function are:
- Domain: [−1, 1]
- Range: [−π/2, π/2]
- Odd Function: sin-1 (-x) = −sin-1 x
What are the properties of arccosine (cos-1 x)?
Some of the properties of inverse of cos function are:
- Domain: [−1, 1]
- Range: [0, π]
- Not Odd: cos-1 (-x) = π − cos-1 x
What are the properties of arccosine (tan-1 x)?
Some of the properties of inverse of tan function are:
- Domain: (−∞,∞)
- Range: [−π/2, π/2]
- Odd Function: tan-1 (-x) = − tan-1 x