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CBSE NOTES CLASS 12 MATHEMATICS CHAPTER 2

INVERSE TRIGONMETRIC FUNCTIONS

Domains and Ranges of Different Trigonometric Functions

Function

Domain

Range

sin

R

[– 1, 1]

cos

R

[– 1, 1]

tan

R   x : x = 2n + 1π2, n ∈ Z

R

cot

R – { x : x = nπ, n ∈ Z}

R

sec

R   x : x = 2n + 1π2, n ∈ Z

R – (-1,1)

cosec

R – { x : x = nπ, n ∈ Z}

R – (-1,1)

Inverse Trigonometric Functions and Domains Principal Range

Functions

Domain

Range (Principal value branches)

y = sin–1 x

[–1,1]

π2,π2

y = cos–1 x

[–1,1]

[0,π]

y = cosec–1 x

R– (–1,1)

π2,π2 0

y = sec–1 x

R– (–1,1)

0,ππ2

y = tan–1 x

R

π2,π2

y =cot-1 x

R

(0, π)

Graphs of Different Trigonometric Functions and Inverse Trigonometric Functions

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Properties of Inverse Trigonometric Functions

  1. (i) sin1 1x= cosec1 x,  x  1 or x   1

    (ii)  cos1 1x= sec1x,  x 1 or x   1

    (iii) tan11x= cot1 x,  x > 0


  2. (i) sin1 x= sin1 x,  x   1, 1

    (ii) tan1 x=  tan1 x,  x  R

    (iii) cosec1  x=  cosec1 x ,  x  1

  3. (i) cos1 (x) = π  cos1 x,  x [ 1, 1]

    (ii) sec1 (x) = π  sec1 x,  |x|  1

    (iii) cot1 x= π  cot1 x,  x  R


  4. (i) sin1 x + cos1 x =π2,  x  1, 1

    (ii) tan1 x + cot1 x =π2,  x R

    (iii) cosec1 x + sec1 x =π2,  x  1


  5. (i) tan1 x + tan1 y = tan1 x + y1-xyxy< 1

    (ii) tan1 x- tan1 y = tan1 x - y1+xy, xy>-1


  6. (i) 2 tan1 x = sin1 2x1+x2,  x1

    (ii) 2 tan1 x = cos1 1-x21+x2,  x 0

    (iii) 2 tan1 x = tan1 2x1-x2, 1 < x < 1