Inverse trigonometric functions

Inverse trigonometric functions are defined as the inverse functions of the basic trigonometric functions, which are sine, cosine, tangent, cotangent, secant and cosecant functions. They are also termed arcus functions, antitrigonometric functions or cyclometric functions. These inverse functions in trigonometry are used to get the angle with any of the trigonometry ratios. The inverse trigonometry functions have major applications in the field of engineering, physics, geometry and navigation.

What Are Inverse Trigonometric Functions?

Inverse trigonometric functions are also called “arc functions” since, for a given value of trigonometric functions, they produce the length of arc needed to obtain that particular value. The inverse trigonometric functions perform the opposite operation of the trigonometric functions such as sine, cosine, tangent, cosecant, secant and cotangent. We know that trigonometric functions are applicable, especially to the right-angle triangle. These six important functions are used to find the angle measure in the right-angle triangle when two sides of the triangle measures are known.

The basic inverse trigonometric formulas are as follows:

Inverse Trig Functions Formulas
Arcsine sin-1(-x) = -sin-1(x), x ∈ [-1, 1]
Arccosine cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]
Arctangent tan-1(-x) = -tan-1(x), x ∈ R
Arccotangent cot-1(-x) = π – cot-1(x), x ∈ R
Arcsecant sec-1(-x) = π -sec-1(x), |x| ≥ 1
Arccosecant cosec-1(-x) = -cosec-1(x), |x| ≥ 1

 

Inverse Trigonometric  Functions Graphs

There are particularly six inverse trig functions for each trigonometry ratio. The inverse of six important trigonometric functions are

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

    Inverse Trigonometric Functions Table

    Let us look at all the inverse trigonometric functions with their notation, definition, domain and range.

    Function Name Notation Definition Domain of  x Range
    Arcsine or inverse sine y = sin-1(x) x=sin y −1 ≤ x ≤ 1
    • − π/2 ≤ y ≤ π/2
    • -90°≤ y ≤ 90°
    Arccosine or inverse cosine y=cos-1(x) x=cos y −1 ≤ x ≤ 1
    • 0 ≤ y ≤ π
    • 0° ≤ y ≤ 180°
    Arctangent or

     

    inverse tangent

    y=tan-1(x) x=tan y For all real numbers
    • − π/2 < y < π/2
    • -90°< y < 90°
    Arccotangent or

     

    inverse cot

    y=cot-1(x) x=cot y For all real numbers
    • 0 < y < π
    • 0° < y < 180°
    Arcsecant or

     

    inverse secant

    y = sec-1(x) x=sec y x ≤ −1 or 1 ≤ x
    • 0≤y<π/2 or π/2<y≤π
    • 0°≤y<90° or 90°<y≤180°
    Arccosecant y=csc-1(x) x=csc y x ≤ −1 or 1 ≤ x
    • −π/2≤y<0 or 0<y≤π/2
    • −90°≤y<0°or 0°<y≤90°

    Inverse Trigonometric Functions Derivatives

    The derivatives of inverse trigonometric functions are first-order derivatives. Let us check out the derivatives of all six inverse functions here.

    Inverse Trig Function dy/dx
    y = sin-1(x) 1/√(1-x2)
    y = cos-1(x) -1/√(1-x2)
    y = tan-1(x) 1/(1+x2)
    y = cot-1(x) -1/(1+x2)
    y = sec-1(x) 1/[|x|√(x2-1)]
    y = csc-1(x) -1/[|x|√(x2-1)]

     

    Properties of Inverse Trigonometric Functions

    Some properties of inverse trigonometric functions are listed below:

    Set 1:

    sin−1(1/x) = cosec−1x, x ≥ 1 or x ≤ −1

    cos−1(1/x) = sec−1x, x ≥ 1 or x ≤ −1

    tan−1(1/x) = cot–1x, x > 0

    Set 2:

    (i) sin–1(–x) = – sin–1 x, x ∈ [– 1, 1]

    (ii) tan–1(–x) = – tan–1 x, x ∈ R

    (iii) cosec–1(–x) = – cosec–1 x, | x | ≥ 1

    Set 3:

    (i) cos–1(–x) = π – cos–1 x, x ∈ [– 1, 1]

    (ii) sec–1(–x) = π – sec–1 x, | x | ≥ 1

    (iii) cot–1(–x) = π – cot–1 x, x ∈ R

    Set 4:

    (i) sin–1 x + cos–1 x = π/2, x ∈ [– 1, 1]

    (ii) tan–1 x + cot–1 x = π/2, x ∈ R

    (iii) cosec–1 x + sec–1 x = π/2, |x| ≥ 1

    Set 5:

    Inverse Trigonometric Functions For Class 12 properties 5

    Set 6:

    Inverse Trigonometric Functions For Class 12 properties 6