# Basic Concepts of Inverse Trigonometric Functions

#### description

• sine, cosine, tangent, cotangent, secant, cosecant function

#### notes

In Class XI, we have studied trigonometric functions, which are defined as follows:

 Trigonometric functions Domain Range sin R [-1,1] cos R [-1.1] tan R- {x:x= (2n+1)π/2,  n∈ Z} R cot R- {x:x= nπ, n∈ Z} R sec R- {x:x= (2n+1) π/2, n∈ Z} R- (-1,1) cosec R- {x:x= nπ, n∈ Z} R- (-1,1)

We already know that, if f: x→y
f(x)= y and f is bijective,
then there exists g: y→x
g(y)= x
g is called "f"^-1
So, with the help of this knowledge we will find the inverse of the trigonometric functions
1)Inverse of sin-
Sin R→[-1,1]
sin x= y

sin: [-π/2, π/2]→[-1,1]

sin^(-1): [-1,1]→[-π/2, π/2], here [-π/2, π/2] is the principle range

sin^(-1)y= x

2) Inverse of cos-
cos: R→[-1,1]

cos: [0, π]→[-1,1]
cos (x)=y is a bijective function
cos^(-1): [-1,1]→[0,π], here [0,π] is the principle range
cos^-y=x
3) Inverse of tan-
tan: "R"- {x:x= (2n+1)π/2,  n∈ "Z"}→"R"

tan: [(-π/2),  (π/2)]→"R"

tan x= y is a bijective function

"tan"^(-1): R→(-pi/2, pi/2), where (-pi/2, pi/2) is the principle range.

tan^(-1)y= x

4) Inverse of cot-
"cot": "R"- {x:x= n pi, n∈"Z"}→R}

cot: (0, pi)→"R"
cot x= y is a bijective function
cot ^(-1):" R"→(0, pi) , where (0, pi) is the principle range.
cot :"R"→ (0, pi)
cot^(-1)y= x
5) Inverse of sec-
sec: "R"-{x:x= (2n+1) pi/2}→"R"

sec: [0, pi]- {pi/2}→"R"- (-1,1)

sec x= y is bijective function

sec^-1: "R"- (-1,1) →[0, pi]- {pi/2}, where [0, pi]- {pi/2} is the principle range

6) Inverse of cosec-
cosec: "R"- {x=n pi, n∈"Z"}→"R"-(-1,1)

cosec: [-pi/2, pi/2] -{0}→"R"-(-1,1)

cosec x= y is bijective function

cosec^-1: "R"-(-1,1)→[-pi/2, pi/2]-{0}, where [-pi/2, pi/2] -{0} is the principle range.

The following table gives the inverse trigonometric function (principal value branches) along with their domains and ranges.

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#### Video Tutorials

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Series 1

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Inverse Trigonometry Functions part 2 (Natural domain Range) [00:04:24]
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