✅ Inverse Trigonometric Formulas ⭐️⭐️⭐️⭐️⭐

Inverse Trigonometric Formulas

Inverse Trigonometric Formulas: Trigonometry is a part of geometry, where we learn about the relationships between angles and sides of a right-angled triangle. In Class 11 and 12 Maths syllabus, you will come across a list of trigonometry formulas, based on the functions and ratios such as, sin, cos and tan. Similarly, we have learned about inverse trigonometry concepts also. The inverse trigonometric functions are written as sin^-1x, cos^-1x, cot^-1 x, tan^-1 x, cosec^-1 x, sec^-1 x. Now, let us get the formulas related to these functions.

What is Inverse Trigonometric Function?

The inverse trigonometric functions are also known as the anti trigonometric functions or sometimes called arcus functions or cyclometric functions. The inverse trigonometric functions of sine, cosine, tangent, cosecant, secant and cotangent are used to find the angle of a triangle from any of the trigonometric functions. It is widely used in many fields like geometry, engineering, physics, etc. But in most of the time, the convention symbol to represent the inverse trigonometric function using arc-prefix like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). To determine the sides of a triangle when the remaining side lengths are known.

Consider, the function y = f(x), and x = g(y) then the inverse function is written as g = f^-1,

This means that if y=f(x), then x = f^-1(y).

Such that f(g(y))=y and g(f(y))=x.

Example of Inverse trigonometric functions: x= sin^-1y

The list of inverse trigonometric functions with domain and range value is given below:

Functions	Domain	Range
Sin-1 x	[-1, 1]	[-π/2, π/2]
Cos-1x	[-1, 1]	[0, π/2]
Tan-1 x	R	(-π/2, π/2)
Cosec-1 x	R-(-1,1)	[-π/2, π/2]
Sec-1 x	R-(-1,1)	[0,π]-{ π/2}
Cot-1 x	R	[-π/2, π/2]-{0}

Inverse Trigonometric Formulas List

To solve the different types of inverse trigonometric functions, inverse trigonometry formulas are derived from some basic properties of trigonometry. The formula list is given below for reference to solve the problems.

S.No	Inverse Trigonometric Formulas
1	sin-1(-x) = -sin-1(x), x ∈ [-1, 1]
2	cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]
3	tan-1(-x) = -tan-1(x), x ∈ R
4	cosec-1(-x) = -cosec-1(x), |x| ≥ 1
5	sec-1(-x) = π -sec-1(x), |x| ≥ 1
6	cot-1(-x) = π – cot-1(x), x ∈ R
7	sin-1x + cos-1x = π/2 , x ∈ [-1, 1]
8	tan-1x + cot-1x = π/2 , x ∈ R
9	sec-1x + cosec-1x = π/2 ,|x| ≥ 1
10	sin-1(1/x) = cosec-1(x), if x ≥ 1 or x ≤ -1
11	cos-1(1/x) = sec-1(x), if x ≥ 1 or x ≤ -1
12	tan-1(1/x) = cot-1(x), x > 0
13	tan-1 x + tan-1 y = tan-1((x+y)/(1-xy)), if the value xy < 1
14	tan-1 x – tan-1 y = tan-1((x-y)/(1+xy)), if the value xy > -1
15	2 tan-1 x = sin-1(2x/(1+x2)), |x| ≤ 1
16	2tan-1 x = cos-1((1-x2)/(1+x2)), x ≥ 0
17	2tan-1 x = tan-1(2x/(1-x2)), -1<x<1
18	3sin-1x = sin-1(3x-4x3)
19	3cos-1x = cos-1(4x3-3x)
20	3tan-1x = tan-1((3x-x3)/(1-3x2))
21	sin(sin-1(x)) = x, -1≤ x ≤1
22	cos(cos-1(x)) = x, -1≤ x ≤1
23	tan(tan-1(x)) = x, – ∞ < x < ∞.
24	cosec(cosec-1(x)) = x, – ∞ < x ≤ 1 or -1 ≤ x < ∞
25	sec(sec-1(x)) = x,- ∞ < x ≤ 1 or 1 ≤ x < ∞
26	cot(cot-1(x)) = x, – ∞ < x < ∞.
27	sin-1(sin θ) = θ, -π/2 ≤ θ ≤π/2
28	cos-1(cos θ) = θ, 0 ≤ θ ≤ π
29	tan-1(tan θ) = θ, -π/2 < θ < π/2
30	cosec-1(cosec θ) = θ, – π/2 ≤ θ < 0 or 0 < θ ≤ π/2
31	sec-1(sec θ) = θ, 0 ≤ θ ≤ π/2 or π/2< θ ≤ π
32	cot-1(cot θ) = θ, 0 < θ < π

The inverse trigonometric formula list helps the students to solve the problems in an easy way by applying those properties to find out the solutions.

What is an Inverse Trigonometric Function?

In geometry, the part that tells us about the relationships existing between the angles and sides of a right-angled triangle is known as trigonometry. It has formulas and identities that offer great help in mathematical and scientific calculations. Along with that trigonometry also has functions and ratios such as sin, cos, and tan. In the same way, we can answer the question of what is an inverse trigonometric function? 

Well, there are inverse trigonometry concepts and functions that are useful. Inverse trigonometry formulas can help you solve any related questions. Just as addition is an inverse of subtraction and multiplication is an inverse of division, in the same way, inverse functions in an inverse trigonometric function. We can call it by different names such as anti-trigonometric functions, arcus functions, and cyclometric functions. The inverse trigonometric functions are as popular as anti trigonometric functions. The inverse functions have the same name as functions but with a prefix “arc” so the inverse of sine will be arcsine, the inverse of cosine will be arccosine, and tangent will be arctangent. So now when next time someone asks you what is an inverse trigonometric function?  You have a lot to say.  

y=sin^-1x

Inverse Trigonometric Functions Formulas

Inverse trigonometric functions formula helps the students to solve the toughest problem easily, all thanks to inverse trigonometry formula. Some of the inverse trigonometric functions formulas are:

1. sin^-1(x) = – sin^-1x
2. cos^-1(x) = π – cos^-1x
3. sin^-1(x) + cos^-1x = π/2

So these were some of the inverse trigonometric functions formulas that you can use while solving trigonometric problems

Table of Inverse Trigonometric Functions

Fun Facts

• Hipparchus, the father of trigonometry compiled the first trigonometry table
• Inverse trigonometric functions were actually introduced early in 1700x by Daniel Bernoulli. 

Solved Examples 

Example 1) Find the value of tan^-1(tan 9π/ 8 )

Solution 1)  tan^-1(tan9π/8)       

= tan^-1tan ( π + π/8)

 = tan^-1 (tan(π/ 8))

=π / 8

Example 2) Find sin (cos^-13/5)

Solution 2) Suppose that, cos^-1 3/5 = x

So, cos x = 3/5

This implies, sin x = sin (cos^-1 3/5) = ⅘

Example 3) Prove the equation “Sin^-1 (-x) = – Sin^-1 (x), x ϵ (-1, 1)”

Solution 3) Let Sin^-1 (-x) = y

Then -x = sin y

x = – sin y

x =sin (-y)

sin^-1 -x = arcsin ( sin(-y))

sin^-1 -x = y

Hence, Sin^-1 (-x) = – Sin^-1 (x), x ϵ (-1, 1)

Example 4) Prove – Cos^-1 (4x³ -3 x) =3 Cos^-1 x , ½ ≤ x ≤ 1

Solution 4)  Let x = Cos ϴ

Where ϴ = Cos-¹ (-x)

LHS = Cos-¹ (-x) (4x³ -3x)

By substituting the value of x, we get

= Cos^-1 (-x) (4 Cos³ϴ – 3 Cosϴ)

Accordingly, we get,

 Cos^-1 (Cos 3ϴ)

= 3ϴ

By substituting the value of ϴ, we get

= 3 Cos^{-1 x}

= RHS

Hence proved

FAQs (Frequently Asked Questions)

Question 1) What are the applications of Inverse Trigonometric Functions?

Answer 1) The inverse trigonometric formula’s major role is to help us in finding out the unknown measurement of an angle of a right angle triangle when any of its two sides are provided. We use the trigonometric function particularly on the basis of which sides are known to us. In other words, if the measurement of the side of the hypotenuse and the side opposite to the angle ϴ are known to us then we use an inverse sine function. In the same way, if we are provided with the measurement of the adjacent side and the opposite side then we use an inverse tangent function for the determination of a right-angle triangle. The inverse trigonometric function extends its hand even to the field of engineering, physics, geometry, and navigation. An inverse trigonometric function can be determined by two methods. The first is to use the trigonometric ratio table and the second is to use scientific calculators.

Question 2) What are Trigonometric Functions?

Answer 2) Trigonometry is the science of measuring triangles. We can refer to trigonometric functions as the functions of an angle of a triangle. In other words, it is these trig functions that define the relationship that exists between the angles and sides of a triangle. 

There are six main trigonometric functions that are given below:

1. Sine (sin)
2. Cosine (cos)
3. Tangent (tan)
4. Secant (sec)
5. Cosecant (cosec)
6. Cotangent (cot)

We use these functions to relate the angles and the sides of a right-angled triangle. Trigonometric functions are important when we are studying triangles. 

Inverse Trigonometric Functions

Inverse trigonometric functions, as a topic of learning, are closely related to the basic trigonometric functions. The domain and the range of the trigonometric functions are converted to the range and domain of the inverse trigonometric functions. In trigonometry, we learn about the relationships between angles and sides in a right-angled triangle. Similarly, we have inverse trigonometry functions. The basic trigonometric functions are sin, cos, tan, cosec, sec, and cot. The inverse trigonometric functions on the other hand are denoted as sin^-1x, cos^-1x, cot^-1 x, tan^-1 x, cosec^-1 x, and sec^-1 x.

Inverse trigonometric functions have all the formulas of the basic trigonometric functions, which include the sum of functions, double and triple of a function. Here we shall try to understand the transformation of the trigonometric formulas to inverse trigonometric formulas.

What are Inverse Trigonometric Functions?

Inverse trigonometric functions are the inverse functions relating to the basic trigonometric functions. The basic trigonometric function of sin θ = x, can be changed to sin^-1 x = θ. Here, x can have values in whole numbers, decimals, fractions, or exponents. For θ = 30° we have θ = sin^-1 (1/2), where θ lies between 0° to 90°. All the trigonometric formulas can be transformed into inverse trigonometric function formulas.

Inverse trigonometric functions are also known as the anti-trigonometric functions/ arcus functions/ cyclometric functions. Inverse trigonometric functions are the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. The inverse trigonometric functions are written using arc-prefix like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). The inverse trigonometric functions are used to find the angle of a triangle from any of the trigonometric functions. It is used in diverse fields like geometry, engineering, physics, etc.

Consider, the function y = f(x), and x = g(y) then the inverse function can be written as g = f^-1,

This means that if y = f(x), then x = f^-1(y).

An example of inverse trigonometric function is x = sin^-1y.

Inverse Trigonometric Formulas

The list of inverse trigonometric formulas has been grouped under the following formulas. These formulas are helpful to convert one function to another, to find the principal angle values of the functions, and to perform numerous arithmetic operations across these inverse trigonometric functions. Further all the basic trigonometric function formulas have been transformed to the inverse trigonometric function formulas and are classified here as the following four sets of formulas.

• Arbitrary Values
• Reciprocal and Complementary functions
• Sum and difference of functions
• Double and triple of a function

Inverse Trigonometric Function Formulas for Arbitrary Values

The inverse trigonometric function formula for arbitrary values is applicable for all the six trigonometric functions. For the inverse trigonometric functions of sine, tangent, cosecant, the negative of the values are translated as the negatives of the function. And for functions of cosecant, secant, cotangent, the negatives of the domain are translated as the subtraction of the function from the π value.

• sin^-1(-x) = -sin^-1x,x ∈ [-1,1]
• tan^-1(-x) = -tan^-1x, x ∈ R
• cosec^-1(-x) = -cosec^-1x, x ∈ R – (-1,1)
• cos^-1(-x) = π – cos^-1x, x ∈ [-1,1]
• sec^-1(-x) = π – sec^-1x, x ∈ R – (-1,1)
• cot^-1(-x) = π – cot^-1x, x ∈ R

Inverse Trigonometric Function 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 can also be expressed in the following forms.

• 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
  tan^-1x = – π + cot^-1 x, x < 0

Inverse Trigonometric Function Formulas for Complementary Functions

The sum of the complementary inverse trigonometric functions results in a right angle. For the same values of x, the sum of complementary inverse trigonometric functions is equal to a right angle. Therefore, complementary functions of sine-cosine, tangent-cotangent, secant-cosecant, sum up to π/2. The complementary functions, sine-cosine, tangent-cotangent, and secant-cosecant can be interpreted as,

• 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), 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 of Inverse Trigonometric Function Formulas

The double of an inverse trigonometric function can be solved to form a single trigonometric function as per the below set of formulas.

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

Triple of Inverse Trigonometric Function Formulas

The triple of the inverse trigonometric functions can be solved to form a single inverse trigonometric function as per the below set of formulas.

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

Let us study the properties of inverse trigonometric functions using their graph, domain, and range in the following sections.

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

Inverse Trigonometric Functions Graph

We can plot the graphs of different inverse trigonometric functions with their range of principal values. Here, we have chosen random values for x in the domain of respective inverse trigonometric functions. We will understand the domain and range of these functions in the following sections.

Arcsine Function

Arcsine function or inverse sine function, also denoted as sin^-1 x, is an inverse of the sine function. The graph of sine inverse function is as given below.

Arccosine Function

Arccosine function or inverse cosine function, also denoted as cos^-1 x, is an inverse of the cosine function. The graph of cos inverse function is as given below.

Arctangent Function

Arctangent function or inverse tangent function, also denoted as tan^-1 x, is an inverse of the tan function. The graph of tan inverse function is as given below.

Arccotangent Function

Arccotangent function or inverse cotangent function, also denoted as cot^-1 x, is an inverse of the cotangent function. The graph of cot inverse function is as given below.

Arcsecant Function

Arcsecant function or inverse secant function, also denoted as sec^-1 x, is an inverse of the secant function. The graph of sec inverse function is as given below.

Inverse Trigonometric Functions Domain and Range

The above graphs help us to compare and understand the functions y = sin^-1x, y = cos^-1x, y = tan^-1x, y = cot^-1x, y = sec^-1x, and y = cosec^-1x. The domain(x value) of the function is presented along the x-axis and the range(y value) of the inverse trigonometric function is presented along the y-axis. The below-given table gives the domain and range of principal values of inverse trigonometric functions.

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 trigonometric functions.

Tips and Tricks on Inverse Trigonometric Functions

Some of the below tips would be helpful in solving and apply the various formulas of inverse trigonometric functions.

• sin^-1(sin x) = sin(sin^-1x) = x, -π/2 ≤ x ≤π/2.
• sin^-1x is different from (sin x)^-1. Also (sin x)^-1 = 1/sinx
• sin^-1x = θ and θ refer to the angle, which is the principal value of this inverse trigonometric function.

Solved Examples on Inverse Trigonometric Functions

Example 1: Find the principal value of cos^-1(-1/2). Solution: Let us assume that, x = cos^-1( -1/2) We can write this as: cos x = -1/2 The range of the principal value of cos^-1 is [0,π ]. Thus, the principal value of cos^-1(-1/2) is 2π/3.
Answer: Hence, the principal value of the cos^-1(-1/2) is 2π/3.

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

Solution: sin^-1 (1/2) – sec^-1(-2) = π/6 – (π – sec^-12)
= π/6 – (π – π/3) (The range of cos⁻¹(x) is [0,π], we can find x using x = cos^-1(1/2) = π/3)
= π/6 – π + π/3
= π/6 + π/3 – π
= π/2 – π
= -π/2
Answer: Hence the value of the given expression is -π/2

Example 3: Find the value of cos^-1 (cos 13π/6).

Solution: cos^-1 (cos 13π/6) = cos^-1 [cos(2π + π/6)]

= cos^-1 [cos π/6]

= π/6

Answer: The value of cos^-1 (cos 13π/6) is π/6.

Inverse Trigonometric Identities

In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inverse trigonometric functions are generally used in fields like geometry, engineering, etc. The representation of inverse trigonometric functions are:

If a = f(b), then the inverse function is

b = f^-1(a) 

Examples of inverse inverse trigonometric functions are sin^-1x, cos^-1x, tan^-1x, etc.

The following table shows some trigonometric functions with their domain and range.

Function	Domain	Range
y = sin-1 x	[-1, 1]	[-π/2, π/2]
y = cos-1 x	[-1, 1]	[0 , π]
y = cosec-1 x	R – (-1,1 )	[-π/2, π/2] – {0}
y = sec-1 x	R – (-1, 1)	[0 , π] – {π/2}
y = tan-1 x	R	(-π/2, π/2)
y = cot-1 x	R	(0 , π)

Properties of Inverse Trigonometric Functions

The following are the properties of inverse trigonometric functions:

Property 1:

1. sin^-1 (1/x) = cosec^-1 x, for x ≥ 1 or x ≤ -1
2. cos^-1 (1/x) = sec^-1 x, for x ≥ 1 or x ≤ -1
3. tan^-1 (1/x) = cot^-1 x, for x > 0

Property 2:

1. sin^-1 (-x) = -sin^-1 x, for x ∈ [-1 , 1]
2. tan^-1 (-x) = -tan^-1 x, for x ∈ R
3. cosec^-1 (-x) = -cosec^-1 x, for |x| ≥ 1

Property 3

1. cos^-1 (-x) = π – cos^-1 x, for x ∈ [-1 , 1]
2. sec^-1 (-x) = π – sec^-1 x, for |x| ≥ 1
3. cot^-1 (-x) = π – cot^-1 x, for x ∈ R

Property 4

1. sin^-1 x + cos^-1 x = π/2, for x ∈ [-1,1]
2. tan^-1 x + cot^-1 x = π/2, for x ∈ R
3. cosec^-1 x + sec^-1 x = π/2 , for |x| ≥ 1

Property 5

1. tan^-1 x + tan^-1 y = tan^-1 ( x + y )/(1 – xy), for xy < 1
2. tan^-1 x – tan^-1 y = tan^-1 (x – y)/(1 + xy), for xy > -1
3. tan^-1 x + tan^-1 y = π + tan^-1 (x + y)/(1 – xy), for xy >1 ; x, y >0

Property 6

1. 2tan^-1 x = sin^-1 (2x)/(1 + x²), for |x| ≤ 1
2. 2tan^-1 x = cos^-1 (1 – x²)/(1 + x²), for x ≥ 0
3. 2tan^-1 x = tan^-1 (2x)/(1 – x²), for -1 < x <1

Identities of Inverse Trigonometric Function

The following are the identities of inverse trigonometric functions:

1. sin^-1 (sin x) = x provided –π/2 ≤ x ≤ π/2
2. cos^-1 (cos x) = x provided 0 ≤ x ≤ π
3. tan^-1 (tan x) = x provided –π/2 < x < π/2
4. sin(sin^-1 x) = x provided -1 ≤ x ≤ 1
5. cos(cos^-1 x) = x provided -1 ≤ x ≤ 1
6. tan(tan^-1 x) = x provided x ∈ R
7. cosec(cosec^-1 x) = x provided -1 ≤ x ≤ ∞ or -∞ < x ≤ 1
8. sec(sec^-1 x) = x provided 1 ≤ x ≤ ∞ or -∞ < x ≤ 1
9. cot(cot^-1 x) = x provided -∞ < x < ∞

Sample Problems

Question 1: Prove sin^-1 x = sec^-1 1/√(1-x²)

Solution: 

Let sin^-1 x = y

⇒ sin y = x , (since sin y = perpendicular/hypotenuse ⇒ cos y = √(1- perpendicular² )/hypotenuse )

⇒ cos y = √(1 – x²), here hypotenuse = 1

⇒ sec y = 1/cos y

⇒ sec y = 1/√(1 – x²)

⇒ y = sec^-1 1/√(1 – x²)

⇒ sin^-1 x = sec^-1 1/√(1 – x²)

Hence, proved.

Question 2: Prove tan^-1 x = cosec^-1 √(1 + x²)/x

Solution:

Let tan^-1 x = y

⇒ tan y = x , perpendicular = x and base = 1

⇒ sin y = x/√(x² + 1) , (since hypotenuse = √(perpendicular² + base² ) )

⇒ cosec y = 1/sin y

⇒ cosec y = √(x² + 1)/x

⇒ y = cosec^-1 √(x² + 1)/x

⇒ tan^-1 x = cosec^-1 √(x² + 1)/x

Hence, proved.

Question 3: Evaluate tan(cos^-1 x)

Solution: 

Let cos^-1 x = y

⇒ cos y = x , base = x and hypotenuse = 1 therefore sin y = √(1 – x²)/1

⇒ tan y = sin y/ cos y

⇒ tan y = √(1 – x²)/x

⇒ y = tan^-1 √(1 – x²)/x

⇒ cos^-1 x = tan^-1 √(1 – x²)/x

Therefore, tan(cos^-1 x) = tan(tan^-1 √(1 – x²)/x ) = √(1 – x²)/x.

Question 4: tan^-1 √(sin x) + cot^-1 √(sin x) = y. Find cos y.

Solution: 

We know that tan^-1 x + cot^-1 x = /2 therefore comparing this identity with the equation given in the question we get y = π/2

Thus, cos y = cos π/2 = 0.

Question 5: tan^-1 (1 – x)/(1 + x) = (1/2)tan^-1 x, x > 0. Solve for x.

Solution: 

tan^-1 (1 – x)/(1 + x) = (1/2)tan^-1 x

⇒ 2tan^-1 (1 – x)/(1 + x) = tan^-1 x     …(1)

We know that, 2tan^-1 x = tan^-1 2x/(1 – x²).

Therefore, LHS of equation (1) can be written as

tan^-1 [ { 2(1 – x)/(1 + x)}/{ 1 – [(1 – x)(1 + x)]²}]

= tan^-1 [ {2(1 – x)(1 + x)} / { (1 + x)² – (1 – x)² }]

= tan^-1 [ 2(1 – x²)/(4x)]

= tan^-1 (1 – x²)/(2x)

Since, LHS = RHS therefore

tan^-1 (1 – x²)/(2x) = tan^-1 x

⇒ (1 – x²)/2x = x

⇒ 1 – x² = 2x²

⇒ 3x² = 1

⇒ x = ± 1/√3

Since, x must be greater than 0 therefore x = 1/√3 is the acceptable answer.

Question 6: Prove tan^-1 √x = (1/2)cos^-1 (1 – x)/(1 + x)

Solution: 

Let tan^-1 √x = y

⇒ tan y = √x

⇒ tan² y = x

Therefore,

RHS = (1/2)cos^-1 ( 1- tan² y)/(1 + tan² y)

= (1/2)cos^-1 (cos² y – sin² y)/(cos² y + sin² y)

= (1/2)cos^-1 (cos² y – sin² y)

= (1/2)cos^-1 (cos 2y)

= (1/2)(2y)

= y

= tan^-1 √x

= LHS

Hence, proved.

Question 7: tan^-1 (2x)/(1 – x²) + cot^-1 (1 – x²)/(2x) = π/2, -1 < x < 1. Solve for x.

Solutions: 

tan^-1 (2x)/(1 – x²) + cot^-1 (1 – x²)/(2x) = π/2

⇒ tan^-1 (2x)/(1 – x²) + tan^-1 (2x)/(1 – x²) = π/2

⇒ 2tan^-1 (2x)/(1 – x²) = ∏/2

⇒ tan^-1 (2x)/(1 – x²) = ∏/4

⇒ (2x)/(1 – x²) = tan ∏/4

⇒ (2x)/(1 – x²) = 1

⇒ 2x = 1 – x²

⇒ x² + 2x -1 = 0

⇒ x = [-2 ± √(2² – 4(1)(-1))] / 2

⇒ x = [-2 ± √8] / 2

⇒ x = -1 ± √2

⇒ x = -1 + √2 or x = -1 – √2

But according to the question x ∈ (-1, 1) therefore for the given equation the solution set is x ∈ ∅.

Question 8: tan^-1 1/(1 + 1.2) + tan^-1 1/(1 + 2.3) + … + tan^-1 1/(1 + n(n + 1)) = tan^-1 x. Solve for x.

Solution:  

tan^-1 1/(1 + 1.2) + tan^-1 1/(1 + 2.3) + … + tan^-1 1/(1 + n(n + 1)) = tan^-1 x  

⇒ tan^-1 (2 – 1)/(1 + 1.2) + tan^-1 (3 – 2)/(1 + 2.3) + … + tan^-1 (n + 1 – n)/(1 + n(n + 1)) = tan^-1 x

⇒ (tan^-1 2 – tan^-1 1) + (tan^-1 3 – tan^-1 2) + … + (tan^-1 (n + 1) – tan^-1 n) = tan^-1 x

⇒ tan^-1 (n + 1) – tan^-1 1 = tan^-1 x

⇒ tan^-1 n/(1 + (n + 1).1) = tan^-1 x

⇒ tan^-1 n/(n + 2) = tan^-1 x

⇒ x = n/(n + 2)

Question 9: If 2tan^-1 (sin x) = tan^-1 (2sec x) then solve for x.

Solution: 

2tan^-1 (sin x) = tan^-1 (2sec x)

⇒ tan^-1 (2sin x)/(1 – sin² x) = tan^-1 (2/cos x)

⇒ (2sin x)/(1 – sin² x) = 2/cos x

⇒ sin x/cos² x = 1/cos x

⇒ sin x cos x = cos² x

⇒ sin x cos x – cos² x = 0

⇒ cos x(sin x – cos x) = 0

⇒ cos x = 0 or sin x – cos x = 0

⇒ cos x = cos π/2 or tan x = tan π/4

⇒ x = π/2 or x = π/4

But at x = π/2 the given equation does not exist hence x = π/4 is the only solution.

Question 10: Prove that cot^-1 [ {√(1 + sin x) + √(1 – sin x)}/{√(1 + sin x) – √(1 – sin x)}] = x/2, x ∈ (0, π/4)

Solution: 

Let x = 2y therefore

LHS = cot^-1 [{√(1+sin 2y) + √(1-sin 2y)}/{√(1+sin 2y) – √(1-sin 2y)}]

= cot^-1 [{√(cos² y + sin² y + 2sin y cos y) + √(cos² y + sin² y – 2sin y cos y)}/{√(cos² y + sin² y + 2sin y cos y) – √(cos² y + sin² y – 2sin y cos y)} ] 

= cot^-1 [{√(cos y + sin y)² + √(cos y – sin y)²} / {√(cos y + sin y)² – √(cos y – sin y)²}] 

= cot^-1 [( cos y + sin y + cos y – sin y )/(cos y + sin y – cos y + sin y)] 

= cot^-1 (2cos y)/(2sin y)

= cot^-1 (cot y)

= y

= x/2.

Some Formulae for Trigonometric Functions

The inverse trigonometric formulae will help the students to solve the problems in an easy way with the application of these properties. Some of them are given here:

Solved Examples for Inverse Trigonometry Formula

What is an Inverse Function?

If y=f(x) and x=g(y) are two functions such that f (g(y)) = y and g (f(y)) = x, then f and y are said to be inverse of each other i.e.,

g = f^-1

If y = f(x) then x = f^-1 (y)

Inverse Trigonometric Formulas

Browse more Topics under Inverse Trigonometric Functions

• Properties of Inverse Trigonometric Functions

Let’s say, if y = sin x , then x = sin^-1 y, similarly for other trigonometric functions. This is one of the inverse trigonometric formulas. Now, y = sin^-1 (x), y ∈ [π/2 , π/2] and x ∈ [-1,1].

• Thus, sin^-1 x has infinitely many values for given x ∈ [-1, 1].
• There is only one value among these values which lies in the interval [π/2, π/2]. This value is called the principal value.

Domain and Range of Inverse Trigonometric Formulas

Solved Examples for You

Question 1. Find the exact value of each expression without a calculator, in [0,2π).

1. sin^-1(−3√2)
2. cos^-1(−2√2)
3. tan^-1√3

Answer:

1. Recall that −3√2 is from the 30−60−90 triangle. The reference angle for sin and 3√2 would be 60∘. Because this is sine and it is negative, it must be in the third or fourth quadrant. The answer is either 4π/3 or 5π/3.
2. −2√2 is from an isosceles right triangle. The reference angle is then 45∘. Because this is cosine and negative, the angle must be in either the second or third quadrant. The answer is either 3π/4 or 5π/4.
3. √3 is also from a 30−60−90 triangle. Tangent is √3 for the reference angle 60∘. Tangent is positive in the first and third quadrants, so the answer would be π/3 or 4π/3.

Notice how each one of these examples yields two answers. This poses a problem when finding a singular inverse for each of the trig functions. Therefore, we need to restrict the domain in which the inverses can be found.

Question 6: Explain the inverse of cos?

Answer: The arccos function happens to be the inverse of the cosine function. It returns the angle whose cosine happens to be a particular number.

Question 7: Explain the inverse of sin?

Answer: The inverse of the sin function happens to be the arcsin function. However, sine itself would not be invertible due to the fact that it’s not injective. Therefore, it’s certainly not bijective (invertible). To obtain the arcsine function, one will have to restrict the domain of sine to [−π2,π2].

Question 8: Who is credited with the invention of inverse trigonometry?

Answer: Daniel Bernoulli considered inverse trigonometric functions early in the 1700s. He used “A. sin” for a number’s inverse sine. Euler came forward with “A t” for the inverse tangent in the year 1736.

Question 9: Explain what is meant by sec theta?

Answer: The reciprocal cosine function happens to be secant: sec (theta)=1/cos(theta). The reciprocal sine function happens to be cosecant, csc(theta)=1/sin(theta). Also, the 3 new functions are cosecant theta, cotangent theta, and secant theta. Secant theta means 1 over x. Cotangent theta means x over y, and cosecant theta refers to 1 over y.

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