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Inverse Trigonometric Formula: Functions, Properties

The Inverse Trigonometric Formula often referred to as arc functions, are defined within specific intervals or under restricted domains. You can find more information about the properties of inverse trigonometric functions here.
authorImageAnchal Singh22 Sept, 2023
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Inverse Trigonometric Formula

Inverse Trigonometric Formula , often referred to as arcus functions, anti trigonometric functions, or cyclometric functions, are mathematical functions that serve as the inverse operations to the basic trigonometric functions, namely sine, cosine, tangent, cotangent, secant, and cosecant. These functions are utilized to find angles corresponding to specific trigonometric ratios. In practical terms, inverse trigonometric functions have significant applications across various disciplines, including engineering, physics, geometry, and navigation.

What Are Inverse Trigonometric Formula?

Inverse trigonometric functions, commonly known as "arc functions" or "arc trigonometric functions," are named so because they determine the length of an arc required to achieve a specific value of a trigonometric function. These functions essentially reverse the operations performed by standard trigonometric functions, including sine, cosine, tangent, cosecant, secant, and cotangent. Trigonometric functions are typically applied in the context of right-angle triangles, and these six fundamental functions play a crucial role in calculating angle measurements within such triangles when the lengths of two sides are known.

Also Check - Sequence and Series Formula

Formulas

The basic inverse trigonometric formulas are as follows:

Inverse Trig Functions Formulas
Arcsine sin -1 (-x) = -sin -1 (x), x ∈ [-1, 1]
Arccosine cos -1 (-x) = π -cos -1 (x), x ∈ [-1, 1]
Arctangent tan -1 (-x) = -tan -1 (x), x ∈ R
Arccotangent cot -1 (-x) = π – cot -1 (x), x ∈ R
Arcsecant sec -1 (-x) = π -sec -1 (x), |x| ≥ 1
Arccosecant cosec -1 (-x) = -cosec -1 (x), |x| ≥ 1

Inverse Trigonometric Functions Table

Function Name Notation Definition Domain of  x Range
Arcsine or inverse sine y = sin -1 (x) x=sin y −1 ≤ x ≤ 1
  • − π/2 ≤ y ≤ π/2
  • -90°≤ y ≤ 90°
Arccosine or inverse cosine y=cos -1 (x) x=cos y −1 ≤ x ≤ 1
  • 0 ≤ y ≤ π
  • 0° ≤ y ≤ 180°
Arctangent or inverse tangent y=tan -1 (x) x=tan y For all real numbers
  • − π/2 < y < π/2
  • -90°< y < 90°
Arccotangent or inverse cot y=cot -1 (x) x=cot y For all real numbers
  • 0 < y < π
  • 0° < y < 180°
Arcsecant or inverse secant y = sec -1 (x) x=sec y x ≤ −1 or 1 ≤ x
  • 0≤y<π/2 or π/2<y≤π
  • 0°≤y<90° or 90°<y≤180°
Arccosecant y=csc -1 (x) x=csc y x ≤ −1 or 1 ≤ x
  • −π/2≤y<0 or 0<y≤π/2
  • −90°≤y<0°or 0°<y≤90°

Inverse Trigonometric Functions Derivatives

The derivatives of inverse trigonometric functions are all first-order derivatives. Let's explore the derivatives of all six inverse functions below.

Inverse Trig Function dy/dx
y = sin -1 (x) 1/√(1-x 2 )
y = cos -1 (x) -1/√(1-x 2 )
y = tan -1 (x) 1/(1+x 2 )
y = cot -1 (x) -1/(1+x 2 )
y = sec -1 (x) 1/[|x|√(x 2 -1)]
y = csc -1 (x) -1/[|x|√(x 2 -1)]

Inverse Trigonometric Functions Properties

The inverse trigonometric functions, often referred to as arc functions, are defined within specific intervals or under restricted domains. You can find more information about the properties of inverse trigonometric functions here.

Also Check - Probability Formula

Fundamental trigonometry concepts encompass essential trigonometric ratios and functions, including sine (sin x), cosine (cos x), tangent (tan x), cosecant (csc x), secant (sec x), and cotangent (cot x).

Also Check - Inverse Matrix Formula

Inverse Trigonometric Functions Problems

Example 1: Determine x for which sin(x) equals 2.

Solution: Given: sin x = 2

x = sin^(-1)(2), which is not feasible.

Thus, there exists no x for which sin x equals 2. Therefore, the domain of sin^(-1)x is limited to values of x between -1 and 1.

Example 2: Calculate sin^(-1)(sin(π/6)).

Solution:

sin^(-1)(sin(π/6)) = π/6 (Utilizing the identity sin^(-1)(sin(x)) = x)

Also Check - Linear Inequalities Formula

Practice Problems

Problem 1: Determine the solution for tan(arcsin(12/13)).

Problem 2: Calculate the value of x such that cos(arccos(1)) equals cos x.

Inverse Trigonometric Formula FAQs

What are inverse trigonometric functions?

Inverse trigonometric functions, denoted as sin^(-1), cos^(-1), tan^(-1), cot^(-1), sec^(-1), and csc^(-1), are functions that provide an angle as an output when given a particular trigonometric ratio as input. They reverse the operations of standard trigonometric functions.

What is the domain of inverse trigonometric functions?

The domain of inverse trigonometric functions depends on the specific function. For arcsin(x) and arccos(x), the domain is [-1, 1], while for arctan(x), it is all real numbers.

What are the principal values of inverse trigonometric functions?

The principal values of inverse trigonometric functions are the values within their primary intervals of definition. For example, the principal values of arcsin(x) and arccos(x) are within the interval [-π/2, π/2], and for arctan(x), they are within (-π/2, π/2).

Can inverse trigonometric functions have multiple solutions?

Yes, inverse trigonometric functions can have multiple solutions, especially when considering periodicity. For example, for arctan(x), there are infinitely many solutions due to the periodic nature of the tangent function.

What are the derivatives of inverse trigonometric functions?

The derivatives of inverse trigonometric functions can be derived using calculus. For example, the derivative of arcsin(x) is 1/√(1 - x^2), the derivative of arccos(x) is -1/√(1 - x^2), and the derivative of arctan(x) is 1/(1 + x^2).
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