Inverse trigonometric

Introduction

Inverse trigonometric functions are closely related to fundamental trigonometric functions as a learning topic. The domain and range of inverse trigonometric functions are translated from the domain and range of trigonometric functions

Inverse trigonometric functions are the inverse functions of basic trigonometric functions. Sin-1 x = θ can be substituted for the basic trigonometric function sin θ = x.  can take the form of whole integers, decimals, fractions, or exponents in this case. We obtain  θ = sin-1 (1/2) for  θ = 30°, where lies between 0° and 90°. Inverse trigonometric function formulas can be made from any trigonometric formula. Inverse trigonometric functions such as arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), and arccot are written with an arc-prefix (x). To get the angle of a triangle from any of the trigonometric functions, utilize the inverse trigonometric functions. It’s employed in a variety of professions, including geometry, engineering, and physics.

Inverse trigonometric function example is x = sin^-1y.

Formulas of Inverse Trigonometric

The following formulas have been combined together to provide a list of inverse trigonometric formulas. These formulas can be used to transform one function to another, as well as to determine the functions’ results. Inverse trigonometric functions can be used to calculate principal angle values and to perform a number of arithmetic operations. Furthermore, all basic trigonometric function formulas have been transformed to inverse trigonometric function formulas and are divided into four categories below.

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

For Arbitrary Values

• 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

Reciprocal and Complementary Function

• 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
• 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]

Sum and Difference of Function

• 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

Double and Triple of Function

• 2sin^-1x = sin^-1(2x.√(1 – x²))
• 2cos^-1x = cos^-1(2x² – 1)
• 2tan^-1x = tan^-1(2x/1 – x²)
• 3sin^-1x = sin^-1(3x – 4x³)
• 3cos^-1x = cos^-1(4x³ – 3x)
• 3tan^-1x = tan^-1(3x – x³/1 – 3x²)

Graph of Inverse Trigonometric Functions

The graphs of several inverse trigonometric functions with their range of principal values can be plotted. In the domain of corresponding inverse trigonometric functions, we have picked random values for x. In the following sections, we’ll learn about the domain and range of these functions.