Inverse Function

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)

Inverse Trigonometric Functions

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 ∈ [π / 2 , π / 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 [π / 2 , π / 2]. This value is called the principal value.

Domain and Range of Inverse Trigonometric FunctionsGraphs of Inverse Trigonometric Functions

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - EquationsCBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Properties of Inverse Trigonometric Functions

Property I

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property II

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property III

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property IV

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property V

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property VI

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property VII

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property VIII

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property IX

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property X

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property XI

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Property XII

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Important Results

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

where Sk denotes the sum of the product of x1,x2,…xn takes k at a time.

Inverse Hyperbolic Functions

If sinh y = x, then y is called the inverse hyperbolic sine of x and it is written as y = sinh-1 x.

Similarly, cosh-1 x, tan h-1 x etc., can be defined,

Domain and Range of Inverse Hyperbolic Functions

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Relation between Inverse Circular Functions and Inverse Hyperbolic Functions

(i) sinh-1x = – i sin-1(ix)

(ii) cosh-1x = – i cos-1 x .

(iii) tanh--1x = – i tan-1(ix)

Important Results

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

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

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

CBSE Class 11 and 12 Maths Notes Inverse T - Functions and T - Equations

Important Points to be Remembered

(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 θ or tan θ can never be a solution of the form (2n + 1) π / 2.

(b) Equation involving coseca or cote can never be a solution of the form θ = nπ.
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