Inverse Trigonometric Formulas
In Trigonometry we learn about the relationships between angles and sides in a rightangled triangle. Similarly, we have inverse trigonometry functions. The basic trigonometric functions are sin, cos, tan, cosec, sec, and cot. The inverse trigonometric functions on the other hand are denoted as sin^{1}x, cos^{1}x, cot^{1} x, tan^{1} x, cosec^{1} x, and sec^{1} x. In this article, let us learn the inverse trigonometric functions with few solved examples.
What Are Inverse Trigonometric Formulas?
Before we learn about inverse trigonometric formulas we must be aware of the inverse trigonometric functions concept. Inverse trigonometric functions are also known as the antitrigonometric functions/ arcus functions/ cyclometric functions. Inverse trigonometric functions are the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. The inverse trigonometric functions are written using arcprefix like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). The inverse trigonometric functions are used to find the angle of a triangle from any of the trigonometric functions. It is used in diverse fields like geometry, engineering, physics, etc.
Consider, the function y = f(x), and x = g(y) then the inverse function can be written as g = f^{1},
This means that if y = f(x), then x = f^{1}(y).
Such that f(g(y)) = y and g(f(y)) = x.
An example of inverse trigonometric function is x = sin^{1}y
The list of inverse trigonometric formulas has been grouped under the following six formulas. These formulas are helpful to convert one function to another, to find the principal angle values of the functions, and to perform numerous arithmetic operations across these inverse trigonometric functions.
Formula 1

Sin^{1}(x) = Sin^{1}x

Tan^{1}(x) = Tan^{1}x

Cosec^{1}(x) = Cosec^{1}x

Cos^{1}(x) = π  Cos^{1}x

Sec^{1}(x) = π  Sec^{1}x

Cot^{1}(x) = π  Cot^{1}x
Formula 2
The reciprocal values of x convert the given inverse trigonometric function into its inverse function. This follows from the trigonometric functions where sin and cosecant are reciprocal to each other, tangent and cotangent are reciprocal to each other, and cos and secant are reciprocal to each other.

Sin^{1}x = Cosec^{1}1/x

Cos^{1}x = Sec^{1}1/x

Tan^{1}x = Cot^{1}1/x
Formula 3
The sum of the complementary inverse trigonometric functions results in a right angle. For the same values of x, the complementary functions have complementary angles as answers, and the sum of it is equal to a right angle.

Sin^{1}x + Cos^{1}x = π/2

Tan^{1}x + Cot^{1}x = π/2

Sec^{1}x + Cosec^{1}x = π/2
Formula 4
The sum and the difference of two inverse trigonometric functions can be combined to form a single inverse function, as per the below set of formulas. These formulas have also been derived from the basic trigonometric ratio formulas.

Sin^{1}x + Sin^{1}y = Sin^{1}(x.√(1y^{2}) + y.√(1x^{2}))

Sin^{1}x  Sin^{1}y = Sin^{1}(x.√(1y^{2})  y.√(1x^{2}))

Cos^{1}x + Cos^{1}y = Cos^{1}(x.y  √(1x^{2}) . √(1y^{2}))

Cos^{1}x  Cos^{1}y = Cos^{1}(x.y + √(1x^{2}) . √(1y^{2}))

Tan^{1}x + Tan^{1}y = Tan^{1}(x + y/ 1  xy)

Tan^{1}x  Tan^{1}y = Tan^{1}(x  y/ 1 + xy)
Formula 5
The double of an inverse trigonometric function can be solved to form a single trigonometric function as per the below set of formulas.

2Sin^{1}x = Sin^{1}(2x.√(1x^{2}))

2Cos^{1}x = Cos^{1}(2x^{2}  1)

2Tan^{1}x = Tan^{1}(2x/ 1x^{2})
Formula 6
The triple of the inverse trigonometric functions can be solved to form a single inverse trigonometric function as per the below set of formulas.

3Sin^{1}x = Sin^{1}(3x  4x^{3})

3Cos^{1}x = Cos^{1}(4x^{3}  3x)

3Tan^{1}x = Tan^{1}(3x  x^{3}/ 1  3x^{2})
Solved Examples Using Inverse Trigonometric Formulas
Example 1: Find the principal value of Cos^{1}(1/2).
Solution:
Let us assume that, x = cos^{1}( 1/2)
We can write this as:
cos x = 1/2
cos x = cos (2π/3).
The range of the principal value of cos^{1} is [0, π ].
Thus, the principal value of cos^{1}( 1/2) is 2π /3.
Answer: Hence the principal value of the inverse of a function cos^{1}(1/2) is 2π/3.
Example 2: Find the value of Sin^{1} (1/2)  Sec^{1}(2).
Solution: Sin^{1} (1/2)  Sec^{1}(2) = π/6  (π  Sec^{1}2)
= π/6  (π  π/3) (The range of cos^{−1}(x) is [0,π], we can find x using x = cos^{1}(1/2) = π/3)
= π/6  π + π/3
= π/6 + π/3  π
= π/2  π
= π/2
Answer: Hence the value of the given expression is π/2
Example 3: Find the value of cos1 (cos 13π/6)
Solution: cos^{1} (cos 13π/6) = cos^{1} [cos(2π + π/6)]
= cos^{1} [cos π/6]
= π/6
Answer: The value of cos1 (cos 13π/6) is π/6
FAQs on Inverse Trigonometric Formulas
What Are Inverse Trigonometric Formulas?
Inverse trigonometric functions are the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions.
How Do you Solve Inverse Trig Functions?
Inverse trig functions could be solved using the list of inverse trigonometric formulas.
What Are the 6 Inverse Trig Functions?
sin^{1}x, cos^{1}x, cot^{1} x, tan^{1}x, cosec^{1}x, and sec^{1}x are the 6 inverse trig functions.
What is the Range and Domain of Inverse Cosine?
The inverse cosine function is written as cos^{1}(x) or arccos(x). Inverse functions swap x and yvalues, thus the range of inverse cosine is 0 to pi and the domain is 1 to 1.