Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse functions of the trigonometric functions. There are inverses of the sine, cosine, cosecant, tangent, cotangent, and secant functions. They are also termed as arcus functions, antitrigonometric functions, or cyclometric functions. These inverse functions in trigonometry are used to get the angle with any of the trigonometry ratios. Let’s discuss each inverse trigonometric function in detail.

arcsine

arcsine function is an inverse of the sine function denoted by sin^-1x. It returns the angle whose sine corresponds to the provided number.

sinθ = (Opposite/Hypotenuse)

=> sin^-1 (Opposite/Hypotenuse) = θ

The theorem of sin inverse is: d/dx sin^-1x = 1/√(1 – x²)

Proof:

sin(θ) = x

now,

f(x) = sin^-1x ..(eq1)

substitute value of sin in eq(1)

f(sin(θ)) = θ

f'(sin(θ))cos(θ) = 1  .. differentiating the equation

we know that,

sin²θ + cos²θ= 1

so,

cos = √(1 – x²)

f'(x) = 1/√(1 – x²)

now,

d/dx sin^-1x = 1/√(1 – x²)

hence proved.

Example:

sin^-1(1/2) = π/6

arccosine

arccosine function is an inverse of the sine function denoted by cos^-1. It returns the angle whose cosine corresponds to the provided number.

cosθ = (Hypotenuse/Adjacent)

=> cos^-1 (Hypotenuse/Adjacent) = θ

The theorem of cos inverse is: d/dx cos^-1(x) = -1/√(1 – x²)

Proof:

cos(θ) = x

θ = arccos(x)

dx = dcos(θ) = −sin(θ)dθ  .. differentiate the equation

now,

we know that,

sin² + cos² = 1

so,

cos = √(1 – x²)

−sin(θ) = −sin(arccos(x)) = -√(1 – x²)

dθ/dx = −1/sin(θ) = -1/√(1 – x²)

so,

dθ/dx cos^-1(x) = -1/√(1 – x²)

hence proved.

Example:

cos^-1(1/2) = π/3

arctangent

arctangent function is an inverse of the tangent function denoted by tan^-1. It returns the angle whose tangent corresponds to the provided number.

tanθ = (Opposite/Adjacent)

=> tan^-1 (Opposite/Adjacent) = θ

The theorem of tan inverse is: d/dx tan^-1(x) = 1/(1 + x²)

Proof:

tan(θ) = x

θ = arctan(x)

we know that,

tan²θ + 1 = sec²θ

dx/dθ = sec²y .. differentiating tan function

dx/dθ = 1+x²   

therefore,

dθ/dx = 1/(1 + x²)

hence proved.

Example:

tan^-1(1) = π/4

Restricting Domains of Functions to Make them Invertible

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.

Domain & Range of Inverse Functions

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 }

Using Inverse Trigonometric Functions with a Calculator

In a scientific calculator, it is possible to find inverse trigonometric functions as well as trigonometric functions. To find trigonometric functions of an angle, enter the chosen angle in degrees or radians. Underneath the calculator, six trigonometric functions will appear sine, cosine,  tangent, cosecant, secant, and cotangent. Similarly to find inverse trigonometric functions in a scientific calculator go to the shift button in the calculator and press it then select any standard trigonometric function such as sine, cosine,  tangent, cosecant, secant, and cotangent. This will enable you to use inverse trigonometric functions. After selecting the trigonometric function just enter your parameter whether in radians or degrees or in the case of inverse functions enter the values that lie within the domain of that particular function and the scientific calculator will solve it.

Inverse Trigonometric Functions

Periodic functions:

Since trigonometric functions are periodic, their inverse functions are varied to write it in the standard format we use the equations provided below.

arcsin(x) = (-1)ⁿarc sin x + πn

arccos(x) = ±arccos x + 2πn

arctan(x) = arctan(x) + πn

arccot(x) = arccot(x) + πn

where n = 0, ±1, ±2, ….

Substituting trigonometric functions in different functions:

• tan(x) = sin(x)/√(1 – sin²(x)) , x ∈ ( -π/2 , π/2 )
• arcsin(a) = arctan(a/√(1 – a²)) , |a| < 1

Derivatives of the inverse trigonometric functions:

d/dx sin^-1(x) = 1/√(1 – x²)

d/dx cos^-1(x) = -1/√(1 – x²)

d/dx tan^-1(x) = 1/(1 + x²)

d/dx cot^-1(x) = -1/(1 + x²)

Properties of Different 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/2) = 1 ⇒ sin^-1(1) = π/2
• sin^-1(sin(π/2)) = π/2
• sin(sin^-1(1)) = 1

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/3) = 1/2 ⇒ cos^-1(1/2) = π/3
• cos^-1(cos(π/3)) = π/3
• 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(1)
• cot^-1(-1) = -cot^-1(1)
• sec^-1(-2) = π  -sec^-1(2)
• cosec^-1(-2) = -cosec^-1(x)

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)