Inverse Trigonometric Formula: Functions, Properties

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.

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Formulas

The basic inverse trigonometric formulas are as follows:

Inverse Trigonometric Functions Table

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 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.

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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).

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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)

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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.