Notes On Basic Concepts - CBSE Class 12 Maths

Trigonometric functions

                 Domain

         Range

      sine

   R

   [-1,1]

      cos

   R

   [-1,1]

      tan

   R- {x:x = (2n+1)n/2, n ∈ Z}

   R

      cot

   R- {x:x = nπ, n ∈ Z}

   R

     secant

   R- {x:x = (2n+1)n/2, n ∈ Z}

   R -(-1,1)

    cosecant

   R- {x:x = nπ, n ∈ Z}

   R - (-1,1)
















Inverse of a function

If function f: X Y such that f(x) = y, is one-one and onto, then there exists a unique function g: Y X defined by g(y) = x, where x X and y=f(x), y ∈ Y.

The function, "g," is called the "inverse of f" and is denoted by f –1.

The concepts of the inverse of a function, and the domain and range of trigonometric functions, we find the inverse trigonometric functions.

Since sin (0) = sin p = 0 The sine function is not a one-one function.

Restricting the domain to [-π/2,π/2], then the range is [-1, 1].

The sine function is a bijection, and sin-1 exists if the domain is [-π/2,π/2] .

Since fof-1(x) = f-1of(x) = x therefore, sin-1(sin x) = sin [sin-1(x)] = x

When the domain of sine is [-π/2,π/2][-3π/2,-π/2],[π/2,3π/2], , then its range is [-1, 1].

Inverse of sine (sin-1 or the arc sine function)

The domain of sin-1 is [-1, 1].

The range of sin-1 is [-π/2,π/2].

The principal value branch of sin-1 is [-π/2,π/2].

If y = sin-1x, then sin y = x.


Sine function:

Sin-1 function:

Since cos (0) = cos (2p) = 1 The cosine function is not a one-one function.

Restricting the domain to [0,p], the range is [-1, 1].

The cos function is a bijection and cos-1 exists.

When the domain of cosine is [-p, 0], [0,p], [p, 2p] etc., the range is [-1, 1].

The inverse of cosine is written as cos-1 or arc cosine function.

The domain of cos-1 is [-1, 1].

The range of cos-1 is [0,p].

The principal value branch of cos-1 is [0,p].

Cosine function:

Cos-1 function:

Similarly, working on the same lines, the other four inverse trigonometric functions can be defined.

The principal value branch of cosec-1 is [-π/2,π/2] - {0}..

cosec-1 :R - (-1, 1) → [-π/2,π/2] - {0}

The principal value branch of sec-1 is [0,p] - {p/2}.
sec-1: R - (-1,1) → [0, π] - {π/2}

The principal value branch of tan-1 is (-π/2,π/2).

tan-1 : R → [-π/2,π/2]

The domain of cot-1 is the set of real numbers.

The range of cot-1 is (0,p).

The principal value branch of cot-1 is (0,p )).

Principal value of an inverse trigonometric function

The value of an inverse trigonometric function, which lies in the range of the principal branch, is called its principal value.

  Inverse trigonometric function

  Principal Value Branch

 sin-1

  [-π/2,π/2]

 cos-1

  [0, p]

 cosec-1

  [-π/2,π/2] - {0}

 sec-1

  [0,p] - {π/2}

 tan-1

  (-π/2,π/2)

 cot-1

  (0,p)

Summary

Trigonometric functions

                 Domain

         Range

      sine

   R

   [-1,1]

      cos

   R

   [-1,1]

      tan

   R- {x:x = (2n+1)n/2, n ∈ Z}

   R

      cot

   R- {x:x = nπ, n ∈ Z}

   R

     secant

   R- {x:x = (2n+1)n/2, n ∈ Z}

   R -(-1,1)

    cosecant

   R- {x:x = nπ, n ∈ Z}

   R - (-1,1)
















Inverse of a function

If function f: X Y such that f(x) = y, is one-one and onto, then there exists a unique function g: Y X defined by g(y) = x, where x X and y=f(x), y ∈ Y.

The function, "g," is called the "inverse of f" and is denoted by f –1.

The concepts of the inverse of a function, and the domain and range of trigonometric functions, we find the inverse trigonometric functions.

Since sin (0) = sin p = 0 The sine function is not a one-one function.

Restricting the domain to [-π/2,π/2], then the range is [-1, 1].

The sine function is a bijection, and sin-1 exists if the domain is [-π/2,π/2] .

Since fof-1(x) = f-1of(x) = x therefore, sin-1(sin x) = sin [sin-1(x)] = x

When the domain of sine is [-π/2,π/2][-3π/2,-π/2],[π/2,3π/2], , then its range is [-1, 1].

Inverse of sine (sin-1 or the arc sine function)

The domain of sin-1 is [-1, 1].

The range of sin-1 is [-π/2,π/2].

The principal value branch of sin-1 is [-π/2,π/2].

If y = sin-1x, then sin y = x.


Sine function:

Sin-1 function:

Since cos (0) = cos (2p) = 1 The cosine function is not a one-one function.

Restricting the domain to [0,p], the range is [-1, 1].

The cos function is a bijection and cos-1 exists.

When the domain of cosine is [-p, 0], [0,p], [p, 2p] etc., the range is [-1, 1].

The inverse of cosine is written as cos-1 or arc cosine function.

The domain of cos-1 is [-1, 1].

The range of cos-1 is [0,p].

The principal value branch of cos-1 is [0,p].

Cosine function:

Cos-1 function:

Similarly, working on the same lines, the other four inverse trigonometric functions can be defined.

The principal value branch of cosec-1 is [-π/2,π/2] - {0}..

cosec-1 :R - (-1, 1) → [-π/2,π/2] - {0}

The principal value branch of sec-1 is [0,p] - {p/2}.
sec-1: R - (-1,1) → [0, π] - {π/2}

The principal value branch of tan-1 is (-π/2,π/2).

tan-1 : R → [-π/2,π/2]

The domain of cot-1 is the set of real numbers.

The range of cot-1 is (0,p).

The principal value branch of cot-1 is (0,p )).

Principal value of an inverse trigonometric function

The value of an inverse trigonometric function, which lies in the range of the principal branch, is called its principal value.

  Inverse trigonometric function

  Principal Value Branch

 sin-1

  [-π/2,π/2]

 cos-1

  [0, p]

 cosec-1

  [-π/2,π/2] - {0}

 sec-1

  [0,p] - {π/2}

 tan-1

  (-π/2,π/2)

 cot-1

  (0,p)

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