Graphs of Inverse Trigonometric Functions-Formulas, Solved Examples, Class 12 Math Chapter 2 Notes Study Material Download free pdf

There are two popular notations used for inverse trigonometric functions:

Adding “arc” as a prefix.

Example: arcsin(x), arccos(x), arctan(x), …

Adding “-1” as superscript.

Example: sin-1(x), cos-1(x), tan-1(x), …

In this article, we will learn about graphs and nature of various inverse functions.

Inverse of Sine Function, y = sin-1(x)

sin-1(x) is the inverse function of sin(x). Its domain is [−1, 1] and its range is [- π/2, π/2]. It intersects the coordinate axis at (0,0). It is an odd function and is strictly increasing in (-1, 1).

Graph of Function

Graph of Sin-1(x). sin-1(x) is the inverse function of sin(x). Its domain is [−1, 1] and its range is [- π/2, π/2]. It intersects the coordinate axis at (0,0). It is an odd function and is strictly increasing in (-1, 1).

Graph of Sin-1(x)

Function Analysis

Domainx∈[−1,1]x∈[−1,1]
Rangey∈[−π2,π2]y∈[2−π​,2π​]
X – Interceptx=0x=0
Y – Intercepty=0y=0
Minima(−1,−π2)(−1,2−π​)
Maxima(1,π2)(1,2π​)
Inflection Points(0,0)(0,0)
ParityOdd Function
MonotonicityIn (-1, 1) strictly increasing

Sample Problems on Inverse Sine Function

Problem 1: Find the principal value of the given equation:

y = sin-1(1/√2)

Solution:

We are given that:

y = sin-1(1/√2)

So we can say that,

sin(y) = (1/√2)

We know that the range of the principal value branch of sin-1(x) is (−π/2, π/2) and sin(π/4) = 1/√2.

So, the principal value of sin-1(1/√2) = π/4.

Problem 2: Find the principal value of the given equation:

y = sin-1(1)

Solution:

We are given that:

y = sin-1(1)

So we can say that,

sin(y) = 1

We know that the range of the principal value branch of sin-1(x) is (−π/2, π/2) and sin(π/2) = 1.

So, the principal value of sin-1(1) = π/2.

Inverse of Cosine Function, y = cos-1(x)

cos-1(x) is the inverse function of cos(x). Its domain is [−1, 1] and its range is [0, π]. It intersects the coordinate axis at (1, π/2). It is neither even nor an odd function and is strictly decreasing in (-1, 1).

Graph of Function

cos-1(x) is the inverse function of cos(x). Its domain is [−1, 1] and its range is [0, π]. It intersects the coordinate axis at (1, π/2). It is neither even nor an odd function and is strictly decreasing in (-1, 1).

Graph of Cos-1(x)

Function Analysis

Domainx∈[−1,1]x∈[−1,1]
Rangey∈[0,π]y∈[0,π]
X – Interceptx=1x=1
Y – Intercepty=π2y=2π
Minima(1,0)(1,0)
Maxima(−1,π)(−1,π)
Inflection Points(0,π2)(0,2π​)
ParityNeither Even Nor Odd
MonotonicityIn (-1, 1) strictly decreasing

Sample Problems on Inverse Cosine Function

Problem 1: Find the principal value of the given equation:

y = cos-1(1/√2)

Solution:

We are given that:

y = cos-1(1/√2)

So we can say that,

cos(y) = (1/√2)

We know that the range of the principal value branch of cos-1(x) is (0, π) and cos(π/4) = 1/√2.

So, the principal value of cos-1(1/√2) = π/4.

Problem 2: Find the principal value of the given equation:

y = cos-1(1)

Solution:

We are given that:

y = cos-1(1)

So we can say that,

cos(y) = 1

We know that the range of the principal value branch of cos-1(x) is (0, π) and cos(0) = 1.

So, the principal value of cos-1(1) = 0.

Inverse of Tangent Function, y = tan-1(x)

tan-1(x) is the inverse function of tan(x). Its domain is ℝ and its range is [-π/2, π/2]. It intersects the coordinate axis at (0, 0). It is an odd function which is strictly increasing in (-∞, ∞).

Graph of Function

tan-1(x) is the inverse function of tan(x). Its domain is ℝ and its range is [-π/2, π/2]. It intersects the coordinate axis at (0, 0). It is an odd function which is strictly increasing in (-∞, ∞).

Graph of tan-1(x)

Function Analysis

Domainx∈Rx∈R
Rangey∈(−π2,π2)y∈(2−π​,2π​)
X – Interceptx=0x=0
Y – Intercepty=0y=0
MinimaThe function does not have any minima points.
MaximaThe function does not have any maxima points.
Inflection Points(0,0)(0,0)
ParityOdd Function
MonotonicityIn (−∞, ∞) strictly Increasing
Asymptotesy=π2 and y=−π2y=2π​ and y=2−π

Sample Problems on Inverse of Tangent Function

Problem 1: Find the principal value of the given equation:

y = tan-1(1)

Solution:

We are given that:

y = tan-1(1)

So we can say that,

tan(y) = (1)

We know that the range of the principal value branch of tan-1(x) is (-π/2, π/2) and tan(π/4) = 1.

So, the principal value of tan-1(1) = π/4.

Problem 2: Find the principal value of the given equation:

y = tan-1(√3)

Solution:

We are given that:

y = tan-1(√3)

So we can say that,

tan(y) = (√3)

We know that the range of the principal value branch of tan-1(x) is (-π/2, π/2) and tan(π/3) = √3.

So, the principal value of tan-1(√3) = π/3.

Inverse of Cosecant Function, y = cosec-1(x)

cosec-1(x) is the inverse function of cosec(x). Its domain is (-∞, -1] U [1, ∞) and its range is [-π/2, 0) U (0, π/2]. It doesn’t intercept the coordinate axis. It is an odd function that is strictly decreasing in its domain.

Graph of Function

cosec-1(x) is the inverse function of cosec(x). Its domain is (-∞, -1] U [1, ∞) and its range is [-π/2, 0) U (0, π/2]. It doesn’t intercept the coordinate axis. It is an odd function that is strictly decreasing in its domain.

Graph of Cosec-1(x)

Function Analysis

Domainx∈(−∞,−1]∪[1,∞)x∈(−∞,−1]∪[1,∞)
Rangey∈[−π2,0)∪(0,π2]y∈[2−π​,0)∪(0,2π​]
X – Interceptϕϕ
Y – Interceptϕϕ
Minima(−1,−π2)(−1,2−π​)
Maxima(1,π2)(1,2π​)
Inflection PointsThe function does not have any inflection points.
ParityOdd Function
MonotonicityIn (1, ∞) it is decreasing and in (-∞, -1) it is decreasing
Asymptotesy = 0

Sample Problems on Inverse Cosecant Function

Problem 1: Find the principal value of the given equation:

y = cosec-1(√2)

Solution:

We are given that:

y = cosec-1(√2)

So we can say that,

cosec(y) = (√2)

We know that the range of the principal value branch of cosec-1(x) is [-π/2, π/2] – {0} and cosec(π/4) = √2.

So, the principal value of cosec-1(√2) = π/4.

Problem 2: Find the principal value of the given equation:

y = cosec-1(1)

Solution:

We are given that:

y = cosec-1(√2)

So we can say that,

cosec(y) = 1

We know that the range of the principal value branch of cosec-1(x) is [-π/2, π/2] – {0} and cosec(π/2) = 1.

So, the principal value of cosec-1(1) = π/2.

Inverse of Secant Function, y = sec-1(x)

sec-1(x) is the inverse function of sec(x). Its domain is (-∞, -1] U [1, ∞) and its range is [0, π/2) U (π/2, π]. It doesn’t intercept the coordinate axis as it is a discontinuous function. It is neither even nor odd function and is strictly increasing in its domain.

Graph of Function

sec-1(x) is the inverse function of sec(x). Its domain is (-∞, -1] U [1, ∞) and its range is [0, π/2) U (π/2, π]. It doesn’t intercept the coordinate axis as it is a discontinuous function. It is neither even nor odd function and is strictly increasing in its domain.

Graph of Sec-1(x)

Function Analysis

Domainx∈(−∞,−1]∪[1,∞)x∈(−∞,−1]∪[1,∞)
Rangey∈[0,π2)∪(π2,π]y∈[0,2π​)∪(2π​,π]
X – Interceptx=1x=1
Y – Interceptϕϕ
Minima(1,0)(1,0)
Maxima(−1,π)(−1,π)
Inflection PointsThe function does not have any inflection points.
ParityNeither Even Nor Odd
MonotonicityIn (1, ∞) it is increasing and in (-∞, -1) it is increasing
Asymptotesy=π2y=2π

Sample Problems on Inverse of Secant Function

Problem 1: Find the principal value of the given equation:

y = sec-1(√2)

Solution:

We are given that:

y = sec-1(√2)

So we can say that,

sec(y) = (√2)

We know that the range of the principal value branch of sec-1(x) is [0, π] – {π/2} and sec(π/4) = √2.

So, the principal value of sec-1(√2) = π/4.

Problem 2: Find the principal value of the given equation:

y = sec-1(1)

Solution:

We are given that:

y = sec-1(1)

So we can say that,

sec(y) = 1

We know that the range of the principal value branch of sec-1(x) is [0, π] – {π/2} and sec(0) = 1.

So, the principal value of sec-1(1) = 0.

Inverse of Cotangent Function, y = cot-1(x)

cot-1(x) is the inverse function of cot(x). Its domain is ℝ and its range is (0, π). It intersects the coordinate axis at (0, π/2). It is neither even nor odd function and is strictly decreasing in its domain.

Graph of Function

cot-1(x) is the inverse function of cot(x). Its domain is ℝ and its range is (0, π). It intersects the coordinate axis at (0, π/2). It is neither even nor odd function and is strictly decreasing in its domain.

Graph of Cot-1(x)

Function Analysis

Domainx∈Rx∈R
Rangey∈(0,π)y∈(0,π)
X – Interceptx = null
Y – Intercepty=π2y=2π
MinimaThe function does not have any minima points.
MaximaThe function does not have any maxima points.
Inflection PointsThe function does not have any inflection points.
ParityNeither Even Nor Odd
MonotonicityIn (-∞, ∞) strictly decreasing
Asymptotesy=0 and y=πy=0 and y=π

Sample Problems on Inverse of Cotangent Function

Problem 1: Find the principal value of the given equation:

y = cot-1(1)

Solution:

We are given that:

y = cot-1(1)

So we can say that,

cot(y) = 1

We know that the range of the principal value branch of cot-1(x) is (-π/2, π/2) and cot(π/4) = 1.

So, the principal value of cot-1(1) = π/4.

Problem 2: Find the principal value of the given equation:

y = cot-1(1/√3)

Solution:

We are given that:

y = cot-1(1/√3)

So we can say that,

cot(y) = (1/√3)

We know that the range of the principal value branch of cot-1(x) is (-π/2, π/2) and cot(π/3) = 1/√3.

So, the principal value of cot-1(1/√3) = π/3.

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.