Class 12 Mathematics (041) · Chapter 2 NotesInverse Trigonometric Functions

A function f has an inverse only if it is one-one and onto. Trigonometric functions, in their natural state, are periodic and thus fail the one-one test over the set of real numbers. For example, the sine function has a range of [-1, 1], but it takes the same value at multiple points like 0, pi, and 2pi. To create an inverse, we restrict the domain to a specific interval where the function is bijective. For sine, this interval is typically [-pi/2, pi/2]. Within these restricted boundaries, we can define a unique inverse function where the domain of the inverse is the range of the original function, and the range of the inverse is the restricted domain of the original.

• f(x) = y implies f^-1(y) = x
• Domain of f^-1 = Range of f
• Range of f^-1 = Restricted Domain of f

The inverse sine function, denoted as sin^-1 or arc sine, is defined with a domain of [-1, 1]. While many intervals could serve as its range, the interval [-pi/2, pi/2] is designated as the principal value branch. Similarly, the cosine function is restricted to the interval [0, pi] to make it bijective. Its inverse, cos^-1, has a domain of [-1, 1] and a principal value range of [0, pi]. Graphs of these functions are reflections of their original trigonometric counterparts across the line y = x, effectively interchanging the x and y coordinates of every point on the curve.

• sin^-1[-1, 1] -> [-pi/2, pi/2]
• cos^-1[-1, 1] -> [0, pi]
• Principal value is the value falling within these specific ranges

The cosecant and secant functions are the reciprocals of sine and cosine, respectively. Their ranges exclude the open interval (-1, 1), which means their inverse functions have a domain of R - (-1, 1). For cosec^-1, the principal value range is [-pi/2, pi/2] excluding {0}, because cosecant is undefined at zero. For sec^-1, the principal value range is [0, pi] excluding {pi/2}, where the secant function is undefined. Understanding these exclusions is vital for correctly identifying the valid outputs of these inverse functions.

• cosec^-1R - (-1, 1) -> [-pi/2, pi/2] - {0}
• sec^-1R - (-1, 1) -> [0, pi] - {pi/2}

The tangent function is defined for all real numbers except odd multiples of pi/2, and its range is the set of all real numbers (R). Consequently, the domain of tan^-1 is R, and its principal value range is the open interval (-pi/2, pi/2). The cotangent function is undefined for integral multiples of pi, with a range of R. Therefore, cot^-1 has a domain of R and a principal value range of the open interval (0, pi). Unlike sine and cosine inverses, these functions are defined for every real number input but have restricted output ranges.

• tan^-1R -> (-pi/2, pi/2)
• cot^-1R -> (0, pi)

Inverse trigonometric functions follow specific algebraic properties within their defined principal branches. For instance, sin(sin^-1 x) = x for x in [-1, 1], and sin^-1(sin x) = x provided x is within the principal range [-pi/2, pi/2]. These properties allow for the simplification of complex expressions. When solving equations or simplifying terms like tan^-1(cos x / (1 - sin x)), we use trigonometric identities to transform the expression into a form that matches the inverse function, allowing them to 'cancel' out within the valid domain.

• sin^-1(sin x) = x only if x is in the principal branch
• Inverse notation sin^-1 x is not the same as (sin x)^-1
• (sin x)^-1 is equal to 1/sin x