Multiple Angle

Inverse Trigonometric Functions: Multiple Angles in Terms of tan-1 (X)

Fundamental trigonometric functions and inverse trigonometric functions are closely connected. The domain and range of an inverse trigonometric function are derived from the fundamental trigonometric function. The links connecting the angle and side of a right-angle triangle are studied in trigonometry.

The Inverse Trigonometric Functions come into play as well. The fundamental trigonometry functions are sin(x), cos(x), tan(x), cosec(x), sec(x), and cot(x). Whereas, inverse trigonometric functions are symbolised by sin-1x, cos-1x, cosec-1 x, cot-1 x, sec-1 x, and tan-1 x.

What are Inverse Trigonometric Functions? 

The Inverse trigonometric functions are the reverse of the fundamental trigonometric functions’ namely; sin, cos, tan, cot, sec, and cosec. The Inverse trigonometric function formulae can be derived from any trigonometric formula. 

Anti-trigonometric, arcus, and cyclometric are all names for inverse trigonometric functions. Inverse trigonometric functions are denoted by the arc as the prefix, such as arcsine, arccosine, arctangent, arccotangent, arcsecant, and arccosecant. Inverse trigonometric functions are used to calculate a triangle’s angle.

Formulas of Inverse Trigonometric

The following formulae have been combined to provide a list of inverse trigonometric formulas. These formulae convert one function to another, calculate the functions’ primary angle values, and execute various arithmetic operations across these inverse trigonometric functions. For each trigonometric ratio, there are six inverse trigonometric functions in particular. Below are the inverses of six primary trigonometric functions:

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

 The following table contains the trigonometric formulas:

What is Inverse tan?

The inverse trigonometric function of the tangent is the inverse tan, which is the reverse of the tan function. The arctan function (pronounced “arctan”) is another name. It’s expressed as “atan x” or “tan-1x” or “arctan x” in maths.

 “tan-1x” means “tan inverse x.”

To understand how the inverse tan function works, examine a few instances: 

• tan 0 = 0 ⇒ 0 = tan-1(0)

• tan π/6 = 1/√3 ⇒ π/6 = tan-1(1/√3)

• tan π/4 = 1⇒ π/4 = tan-1(1)

Formulas of Inverse Tan 

Below are some inverse tan properties and formulas that will help solve the problems of multiple-angle in Terms of tan-1(X).

• tan (tan-1x) = x, for all real numbers x.

• tan-1(tan x) = x, only when x ∈ R – {x: x = (2n + 1) (π/2), where n ∈ Z}.
  i.e., tan-1(tan x) = x only when x is NOT an odd multiple of π/2. Otherwise, tan-1(tan x) is undefined.

• tan-1(-x) = -tan-1x, for all x ∈ R.

• tan-1(1/x) = cot-1x, when x > 0.

• tan-1(1/x)=π-cot-1(x) when x<0

• tan-1x + cot-1x = π/2, when x ∈ R.

• If XY < 1, then tan-1x + tan-1y = tan-1[(x + y)/(1 – XY)].
  Or if XY > -1, then tan-1x – tan-1y = tan-1[(x – y)/(1 + XY)].

• Formulas for 2tan-1x.
  2tan-1x = sin-1(2x / (1+x2)), when |x| ≤ 1
  2tan-1x = cos-1((1-x2) / (1+x2)), when x ≥ 0
  2tan-1x = tan-1(2x / (1-x2)), when -1 < x < 1

Domain and Range of Inverse Tan to Determine Multiple Angles in Terms of Tan(-1)(X)

The tan function can be one-one if its domain is reduced to one of the intervals (-3π/2, -π/2), (-π/2, π/2), (π/2, 3π/2), and so on. In each of these intervals, we get a division of the inverse tan. However, to make it one-one, a domain of the tan function is normally constrained to (-π/2, π/2). Hence, x: (-π/2, π/2). 

The tan function’s domain and range are the domain and range of its inverse tan function, respectively. In other words, arctan x (or) tan-1x: R → (-π/2, π/2). Therefore,

• R is the domain of tan inverse x

• Tan inverse x has a range of (-π/2, π/2)

Example 1. Find the values of the following: a) tan (tan-1 2)   b) tan-1(tan 3π/2).

Solution:a) We know that tan (tan-1 x) = x for all x.

Therefore, tan (tan-1 (2)) = 2.

b) We know that tan-1(tan x) is NOT defined when x is an odd multiple of π/2.

Here 3π/2 is an odd multiple of π/2.

So, tan-1(tan 3π/2) is NOT defined.

2. Evaluate tan (tan-1(– 4)). 

Solution:  Since tan (tan-1x) = x, ∀ x ∈ R, tan (tan-1(– 4) = – 4.

Conclusion

An inverse function will allow a person to perform the opposite operation as the original function. For example, the inverse function of addition is also known as subtraction because it reverses what happened in the addition problem. All of the formulae for fundamental trigonometric functions apply to inverse trigonometric functions, including the total of functions, double of a function, and triple of a function. 

These inverse functions are used to calculate the angle using any trigonometry ratio in trigonometry. Like Tan Inverse, every part of inverse trigonometry holds a significant value. Therefore, the inverse trigonometry functions have substantial applications in engineering, physics, geometry, and navigation.