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Inverse Trigonometric Functions

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Inverse trigonometric functions are the inverse functions of basic trigonometric functions. They are inverses of the sine, cosine, cosecant, tangent, cotangent, and secant functions. They are also termed arcus functions, antitrigonometric functions, or cyclometric functions. We write inverse trigonometric functions as sin-1 x, cos-1 x, tan-1 x, cot-1 x, cosec-1 x, and sec-1 x which are inverse functions of sin x, cos x, tan x, cot x, cosec x, and sec x respectively.

Inverse trigonometric functions have all the formulas that are included in the trigonometric functions such as sum and differences formulas, double angle formulas, triple angle formulas, and others. Let’s discuss each inverse trigonometric function in detail in this article.

What are Inverse Trigonometric Functions?

Inverse trigonometric functions are the trigonometric functions that are used for finding the inverse of the basic trigonometric functions. They are also called “Arc Functions” as for the given value trigonometric functions, they find the length of arc needed to obtain that particular value.

They perform the inverse operation to basic trigonometric functions i.e.

If sin (x) = a

x = sin-1(a)

Also, In general, sin-1[sin(x)] = x

Inverse Trigonometric Function Formulas

There are various inverse trigonometric formulas used some of the important inverse trigonometric formulas are,

Formulas for Negative Functions

The table below discusses the inverse trigonometric function formulas for negative functions.

Inverse Trig Functions

Formulas

Arcsinesin-1(-x) = -sin-1(x), x ∈ [-1, 1]
Arccosinecos-1(-x) = π -cos-1(x), x ∈ [-1, 1]
Arctangenttan-1(-x) = -tan-1(x), x ∈ R
Arccotangentcot-1(-x) = π – cot-1(x), x ∈ R
Arcsecantsec-1(-x) = π -sec-1(x), |x| ≥ 1
Arccosecantcosec-1(-x) = -cosec-1(x), |x| ≥ 1

Formulas for Reciprocal Functions

The inverse trigonometric function for reciprocal values of x converts the given inverse trigonometric function into its reciprocal function. This follows from the trigonometric functions where sin and cosecant are reciprocal to each other, tangent and cotangent are reciprocal to each other, and cos and secant are reciprocal to each other.

The inverse trigonometric formula of inverse sine, inverse cosine, and inverse tangent are expressed as,

  • sin-1x = cosec-11/x, x ∈ R – (-1,1)
  • cos-1x = sec-11/x, x ∈ R – (-1,1)
  • tan-1x = cot-11/x, x > 0

Formulas for Complementary Functions

Complementary functions are the functions whose addition results in right angles. Thus, the sum of the complementary inverse trigonometric functions also results in a right angle. 

The complementary function pairs are, sine-cosine, tangent-cotangent, and secant-cosecant i.e. for a similar function the sum of these functions results in the right angle. Various inverse trigonometric complementary functions are,

  • sin-1x + cos-1x = π/2, x ∈ [-1,1]
  • tan-1x + cot-1x = π/2, x ∈ R
  • sec-1x + cosec-1x = π/2, x ∈ R – [-1,1]

Sum and Difference of Inverse Trigonometric Function Formulas

The sum and the difference of two inverse trigonometric functions can be combined to form a single inverse function, as per the below set of formulas. The sum and the difference of the inverse trigonometric functions have been derived from the trigonometric function formulas of sin(A + B), cos(A + B), and tan(A + B). These inverse trigonometric function formulas can be used to further derive the double and triple function formulas.

  • sin-1x + sin-1y = sin-1(x.(1 – y2) + y√(1 – x2))
  • sin-1x – sin-1y = sin-1(x.(1 – y2) – y√(1 – x2))
  • cos-1x + cos-1y = cos-1(xy – (1 – x2).√(1 – y2))
  • cos-1x – cos-1y = cos-1(xy + (1 – x2).√(1 – y2))
  • tan-1x + tan-1y = tan-1(x + y)/(1 – xy), if xy < 1
  • tan-1x + tan-1y = tan-1(x – y)/(1 + xy), if xy > – 1

Double Inverse Trigonometric Function Formulas

The double inverse trigonometric function formulas are the formulas that give the values of the double angle in the inverse trigonometric function. Some important double inverse trigonometric function formulas are,

  • 2sin-1x = sin-1(2x.√(1 – x2))
  • 2cos-1x = cos-1(2x2 – 1)
  • 2tan-1x = tan-1(2x/1 – x2)

These formulas are derived using the basic double-angle formulas of the trigonometric function.

Triple of Inverse Trigonometric Function Formulas

The triple inverse trigonometric function formulas are the formulas that give the values of the triple angle in the inverse trigonometric function. Some important triple inverse trigonometric function formulas are,

  • 3sin-1x = sin-1(3x – 4x3)
  • 3cos-1x = cos-1(4x3 – 3x)
  • 3tan-1x = tan-1(3x – x3/1 – 3x2)

These formulas are derived using the basic triple-angle formulas of the trigonometric function.

Learn more, Inverse Trigonometric Identities

Inverse Trigonometric Functions Types

There are a total of six Inverse Trigonometric Functions that are,

  • Arcsine
  • Arccosine
  • Arctangent
  • Arccotangent
  • Arcsecant
  • Arccosecant

Let’s learn about each 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.

sin θ = (Opposite/Hypotenuse)

=> sin-1 (Opposite/Hypotenuse) = θ

Example: sin-1(1/2) = π/6

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.

Arcsine Graph

 

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.

cos θ = (Hypotenuse/Adjacent)

=> cos-1 (Hypotenuse/Adjacent) = θ

Example: cos-1(1/2) = π/3

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.

Arccosine Graph

 

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.

tan θ = (Opposite/Adjacent)

=> tan-1 (Opposite/Adjacent) = θ

Example: tan-1(1) = π/4

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.

Arctangent Graph

 

Arccotangent

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

cot θ = (Adjacent/Opposite)

=> cot-1 (Adjacent/opposite) = θ

Example: cot-1(1) = π/4

Arccotangent Graph

 

Arcsecant

Arcsecant function is an inverse of the secant function denoted by sec-1. It returns the angle whose secant corresponds to the provided number.

sec θ = (Base/Hypotenuse)

=> sec-1 (Base/Hypotenuse) = θ

Example: sec-1(2) = π/3

Arcsecant Graph

 

Arccosecant

Arccosecant function is an inverse of the cosecant function denoted by cosec-1. It returns the angle whose cosecant corresponds to the provided number.

cosec θ = (Adjacent/Hypotenuse)

=> Cossec-1 (Adjacent/Hypotenuse) = θ

Example: Cosec-1(2) = π/6

Arccosecant Graph

 

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, and cosec so that they are invertible.

Domain & Range of Inverse Functions

The domain and the range of the inverse trigonometric function are added in the table below.

Function

Domain

Range

sin-1 x          [ -1, 1 ]     [ −π/2, π/2 ]
cos-1 x          [ -1, 1 ]         [ 0, π ]
tan-1 x              R     [ −π/2, π/2 ]
cot-1 x              R         [ 0, π ]
sec-1 x  ( -∞ , -1 ] U [ 1,∞ )      [ 0, π ] − { π/2 }
cosec-1 x  ( -∞ , -1 ] U [ 1, ∞ )[ −π/2 , π/2 ] – { 0 }

Derivatives of Inverse Trigonometric Functions

We can find the differentiation of different inverse trigonometric functions using differentiation formulas. The following table gives the result of the differentiation of inverse trig functions.

Inverse Trigonometric Functions

dy/dx

y = sin-1(x), x ≠ -1, +1

1/√(1-x2)

y = cos-1(x), x ≠ -1, +1

-1/√(1-x2)

y = tan-1(x), x ≠ -i, +i

1/(1+x2)

y = cot-1(x), x ≠ -i, +i

-1/(1+x2)

y = sec-1(x), |x| > 1

1/[|x|√(x2-1)]

y = cosec-1(x), |x| > 1

-1/[|x|√(x2-1)]

Integrals of Inverse Trigonometric Functions

Here are the integral formulas of inverse trigonometric functions. To see how to derive each one of them,

Inverse Trigonometric FunctionIntegral
∫ sin-1x dxx sin-1x + √(1 – x²) + C
∫ cos-1x dxx cos-1x – √(1 – x²) + C
∫ tan-1x dxx tan-1x – (1/2) ln |1 + x²| + C
∫ csc-1x dxx csc-1x + ln |x + √(x² – 1)| + C
∫ sec-1x dxx sec-1x – ln |x + √(x² – 1)| + C
∫ cot-1x dxx cot-1x + (1/2) ln |1 + x²| + C

Read More,

Solved Examples on Inverse Trigonometric Functions

Example 1: Find x, if sin(x) = 1/2

Solution: 

Given, sin x = 1/2

Using inverse trigonometric function formulas,

x = sin-1(1/2)

Using Trigonometric Table

x = sin-1[sin(π/6)]

x = π/6

Thus, the required value of x is π/6

Example 2: Find the value of cos-1(1/2) – sec-1(-2).

Solution: 

cos-1 (1/2) – sec-1(-2) 

= π/3 – (π – sec-12)

Using Trigonometric Table

= π/3 – (π – π/3)

= π/3 – π + π/3

= π/3 + π/3 – π

= 2π/3 – π

= -π/3

Thus, the required value of the given expression is -π/3

Example 3: Find cos [cos-1 (11/15)]

Solution:

= cos [cos-1 (11/15)]

We know that, cos [cos-1 (x)] = x

= 11/15

Example 4: Find the value of cot-1(1) + sin-1(-1/2) + sin-1(1/2).

Solution:

Using inverse trigonometric functions formulas,

= cot-1(1) + sin-1(-1/2) + sin-1(1/2)

= cot-1(1) – sin-1(1/2) + sin-1(1/2)

= cot-1(1)

= π/4

Thus, the required solution is π/4

Example 5: Simplify sin(tan-1x)

Solution:

Taking sin2 (θ) + cos2 (θ) = 1

dividing both sides by sin2(θ)

1 + cot2 (θ) = cosec2 (θ)

1 + 1/tan2 (θ) = 1/sin2 (θ)

(tan2 (θ) + 1)/tan2 (θ) = 1/sin2 (θ)

Taking the inverse

sin2 (θ) = tan2 (θ)/(tan2 (θ) + 1)…(1)

taking θ = tan-1(x) In eq (1)

sin2 (tan-1(x)) = tan2 (tan-1(x))/(tan2 (tan-1(x)) + 1)

We know that, tan (tan-1(x)) = x

sin2 (tan-1(x)) = x2 / x2 + 1

Taking square root on both sides,

sin (tan-1(x)) = x / √(x2 + 1)

FAQs on Inverse Trigonometric Functions

Q1: How to Find the Inverse of Trigonometric Functions?

Answer:

Inverse Trigonometric functions can easily be found using the basic trigonometric function as, If sin x = y then the inverse sin function sin-1 y can easily be calculated as, 

x = sin-1 y

In the same way, we can also calculate the other inverse trigonometric functions

Q2: How to use Inverse Trigonometric Functions?

Answer:

Inverse Trigonometric functions are used to find the inverse of the basic trigonometric functions they are widely used to solve complex problems of Trigonometry, SHM, and others.

Q3: What are Inverse Trigonometric Function Identities for Reciprocal Functions?

Answer:

The inverse trigonometric formula of reciprocal functions, inverse sine, inverse cosine, and inverse tangent are,

  • sin-1x = cosec-11/x, x ∈ R – (-1,1)
  • cos-1x = sec-11/x, x ∈ R – (-1,1)
  • tan-1x = cot-11/x, x > 0

Q4: How many Trigonometric Identities are there?

Answer:

There are three basic trigonometric identities,

  • sin2 θ + cos2 θ = 1
  • 1+tan2 θ  = sec2 θ
  • cosec2 θ = 1 + cot2 θ

Q5: What are the 6 Inverse Trigonometric Functions?

Answer:

The six inverse trigonometric functions are,

  • sin-1 x
  • cos-1 x
  • tan-1 x
  • cot-1 x
  • cosec-1 x
  • sec-1 x

Q6: What is the Range and Domain of Inverse Sine Trigonometric Function?

Answer:

We write the inverse sine function as sin-1(x) or arcsin (x). The domain and range of Inverse Sine Function are,

  • Domain of Inverse Sine Trigonometric Function: [-1, 1]
  • Range of Inverse Sine Trigonometric Function: [0, π]


Last Updated : 24 Jan, 2024
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