INVERSE TRIGONOMETRIC FUNCTIONS

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

Inverse trigonometric functions formulas are mostly the same as the basic trigonometric functions, which include the sum of functions, double and triple of a function.

Today we shall try to understand the transformation of the trigonometric formulas into inverse trigonometric formulas.

What are Inverse Trigonometric Functions?

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

The Inverse trigonometric functions are also called anti-trigonometric functions or arcus functions or cyclometric functions. Inverse trigonometric functions are actually the inverse functions of the basic trigonometric functions which are sin, cos, tan, cot, sec, and cosec functions. The inverse trigonometric functions are widely written using arc-prefixes like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). The inverse trigonometric functions are also used to find the angle of a triangle same as the trigonometric functions. It is used in diverse fields like geometry, physics, engineering, defence, etc.

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

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

For example, the inverse trigonometric function of sin(x)=y is x = sin-¹y.

Inverse Trigonometric Formulas

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

  The Arbitrary Values

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

Inverse Trigonometric Function Formulas for Arbitrary Values

The inverse trigonometric function formula for arbitrary values is also applicable for all the six basic trigonometric functions. For the inverse trigonometric functions of sin, tan, cosec, the negative of the values are translated as the negatives of the function. And similarly for functions of cosec, sec, cot, 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 the reciprocal function. This is actually from the trigonometric functions where sin and cosec are reciprocal to each other, tan and cot are reciprocal to each other, and cos and sec are reciprocal to each other.

The inverse trigonometric formula of inverse sin, inverse cos, and inverse tan 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 trigonometric functions results as right angle. For the same values of x, the sum of the complementary inverse trigonometric functions is equal to a right angle. Therefore, complementary functions of sin-cos, tan-cot, sec-cosec, sum up to π/2. The complementary functions, sin-cos, tan-cot, and sec-cosec 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]

The 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 given below in the set of formulas. The sum and the difference of the inverse trigonometric functions have been derived below 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 inverse 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

Tips and Tricks on Inverse Trigonometric Functions-

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

• 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.

CONCLUSION :

The Inverse trigonometric functions are also known as “Arc Functions” since, for a given value of trigonometric functions, the arch function produces the length of arc needed to obtain that particular value. The inverse trigonometric functions perform the exact opposite operation of the trigonometric functions such as sin, cos, tan, cosec, sec, and cot.