These notes for CBSE Class 12 Mathematics, Chapter 2, cover Inverse Trigonometric Functions. They define inverse functions, explaining that if y = f(x), then x = f-1(y). The notes detail the principal value of inverse trigonometric functions like sin-1(x), specifying its range as [-?/2, ?/2] and domain as [-1, 1]. Key topics include the domain and range of various inverse trigonometric functions, their graphs, and a comprehensive list of properties (Property I to XII). The document also touches upon important results and defines trigonometric equations and their solutions. It distinguishes between principal solutions (the least value satisfying the equation) and general solutions (all possible solutions incorporating periodicity). Crucial points for solving trigonometric equations, such as avoiding squaring without verification and not cancelling common factors, are highlighted. These notes are ideal for students preparing for their Class 12 Maths exams.
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If y = f(x) and x = g(y) are two functions such that f (g(y)) = y and g (f(y)) = x, then f and y are said to be inverse of each other i.e., g = f-1 IF y = f(x), then x = f-1(y)
If y = sin X-1, then x = sin-1 y, similarly for other trigonometric functions. This is called inverse trigonometric function . Now, y = sin-1(x), y ? [? / 2 , ? / 2] and x ? [-1,1]. (i) Thus, sin-1x has infinitely many values for given x ? [-1, 1]. (ii) There is only one value among these values which lies in the interval [? / 2 , ? / 2]. This value is called the principal value.
where Sk denotes the sum of the product of x1,x2,…xn takes k at a time.
Inverse
Trigonometric Equation
An equation involving one or more trigonometrical ratios of unknown angle is called a
trigonometric equation .
Solution/Roots of a Trigonometric Equation
A value of the unknown angle which satisfies the given equation, is called a solution or root of the equation.
The trigonometric equation may have infinite number of solutions.
(i) Principal Solution – The least value of unknown angle which satisfies the given equation, is called a principal solution of trigonometric equation.
(ii) General Solution – We know that, trigonometric function are periodic and solution of trigonometric equations can be generalised with the help of the periodicity of the trigonometric functions.
The solution consisting of all possible solutions of a trigonometric equation is called its general solution.
Important Results
(i) While solving an equation, we have to square it, sometimes the resulting roots does not satisfy the original equation. (ii) Do not cancel common factors involving the unknown angle on LHS and RHS.Because it may be the solution of given equation. (iii) (a) Equation involving sec ? or tan ? can never be a solution of the form (2n + 1) ? / 2. (b) Equation involving coseca or cote can never be a solution of the form ? = n?.
Inverse trigonometric functions are functions that reverse the action of trigonometric functions. If y = sin(x), then x = sin-1(y).
The principal value is the unique value of the inverse trigonometric function that lies within a specified range, for example, for sin-1(x), it lies in the interval [-?/2, ?/2].
The domain of sin-1(x) is [-1, 1] and its principal value range is [-?/2, ?/2].
A trigonometric equation is an equation that involves one or more trigonometric ratios of an unknown angle.
The principal solution is the least value of the unknown angle that satisfies the equation, while the general solution includes all possible solutions, considering the periodicity of trigonometric functions.
Avoid squaring the equation without checking for extraneous roots, and do not cancel common factors involving the unknown angle as they might be solutions.
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