Introduction

Inverse trigonometric functions can be defined as the inverse of certain functions in trigonometry. These are basically the inverse functions of cosine, sine, tangent, secant, cosecant, and cotangent functions. Sometimes inverse trigonometric functions are also referred to as anti trigonometric functions, cyclometric functions, and or arcus functions. These specific inverse functions are used for getting the angle of any king of trigonometric ratio. Inverse trigonometric functions are a vital concept in mathematics and have applications in several other fields including navigation, geometry, engineering, and physics. 

Concepts of Inverse Trigonometric Functions 

Inverse trigonometric functions are basically Arc functions. The inverse trigonometric functions are called by this name due to the fact that a particular value produced by a specific function, in turn, produces an arc length which is vital for obtaining a particular value. From the name itself, it is clear that the main operation of an inverse trigonometric function is to give the inverse or opposite of a particular function. These operations can be of several types including cosine, sine, tangent, secant, cosecant, and cotangent. It is a well-known fact that trigonometric functions are mostly used in the case of a right-angled triangle. The 6 angles mentioned earlier are mostly used for measuring angles within a right-angled triangle when the length of 2 sides of the mentioned triangle is given. 

Inverse Trigonometric Functions Formulas

Throughout business mathematics, we come across many inverse trigonometric formulas that are often used while working out different problems in a number of fields such as navigation, geometry, engineering, and physics. Some of these formulas have been discussed in the following.

  • The sin-1x function is defined only if -1 ≤ x ≤1
  • In this context, if α is taken as the principal value of sin-1x then -π/2 ≤ α ≤ π/2
  • The cos-1x function is defined only if -1 ≤ x ≤1
  • In this context, if α is taken as the principal value of cos-1x then 0 ≤ α ≤ π
  • The tan-1x function is defined for a particular real value of x that is only if – ∞ < x < ∞
  • In this context, if α is taken as the principal value of tan-1x then -π/2 < α < π/2
  • The cot-1x function is defined for a particular real value of x that is only if – ∞ < x < ∞
  • In this context, if α is taken as the principle value of cot-1x then 0 < α < π
  • The sec-1x function is defined only if 1 ≤ |x|
  • In this context, if α is taken as the principal value of sec-1x then 0 ≤ α ≤ π and α≠ π/2
  • The cosec-1x function is defined only if 1 ≤ |x|
  • In this context, if α is taken as the principal value of cosec-1x then -π/2 ≤ α ≤ π/2 and α≠ 0
  • Some other inverse trigonometric circular functions have been outlined in the following

Inverse Trigonometric Functions

Formulas

Arccosine

cos-1 (-x) = π – cos-1x where x belongs to [-1, 1]

Arccotangent

cot-1(-x) = π – cot-1x where x belongs to R

Arccosecant

cosec-1(-x) = – cosec-1x where |x|≥ 1

Arcsine

sin-1(-x) = – sin-1x where x belongs to [-1, 1]

Arctangent

tan-1(-x) = – tan-1x where x belongs to R

Arcsecant

sec-1(-x) = π – sec-1x where |x|≥ 1

Differentiation of Inverse Trigonometric Functions

Generally first order derivatives of inverses trigonometric functions are needed for carrying out different sums and these have been outlined in the following table

Inverse Trigonometric Functions

dy / dx

y = cos-1x

-1 /√ (1- x2)

y = cot-1x

-1 / (1+x2)

y = cosec-1x

-1 / [|x| √( x2 -1)]

y = sin-1x

1 /√ (1- x2)

y = tan-1x

1 / (1+x2)

y = sec-1x

1 / [|x| √( x2 -1)]

In the above table, dy / dx refers to the derivative of y with respect to x.

Conclusion

The main subject on which this article has been written is business mathematics. In this subject, the selected topic is Inverse trigonometric functions. In this article, firstly the definition of an inverse trigonometric function has been properly analysed.  Various fields in which this particular function is generally used have been mentioned here. Next, different inverse trigonometric formulas that are generally utilised by mathematicians and physicians have been mentioned. Lastly, the differentiations or derivatives of different inverse trigonometric functions have been discussed.