INVERSE TRIGONOMETRIC

FUNCTIONS

xyf yxf 1

DEFINITION: If is a one-to-one function with domain A and range B, then its inverse is the function with domain B and range A defined by

f1f

For a function to have an inverse, it must be one-to-one. Since the trigonometric functions are not one-to-one, they do not have inverses. It is possible, however, to restrict the domains of the trigonometric functions in such a way that the resulting functions are one-to-one.

Inverse trigonometric functions are defined as follows:

.1x1 where ,2

y2

and xy sin if only and if xsiny 1

o 2

1

1

x

y

2

x

y

2

2

1 1

2 ,

2:R

1 ,1:D

y = sin-1 x

y = sin x 1 ,1:R

2 ,

2:D

.1x1 where ,y0 and xycos if only and if xcosy 1

,0 :R

1 ,1:D

y = cos-1 x

o

2

1

1

x

y

2

2

x

y

2

1 1

1 1,-:R

,0:D

y = cos x

.x nos. real all for ,2

y2

and xytan if only and if xtany 1

2 ,

2:R

,:D

y = tan-1 x

o

2

1

1

x

y

2

x

y

2

1 1

2

y = tan x

,:R

2 ,

2:D

.x nos. real all for ,y0 and xycot if only and if xcoty 1

,0:R

,:D

y = cot-1 x

o

2

1

1

x

y

2

y = cot x

,:R

,0:D

x

y

2

1 1

.1x -1,x ,2

y ,y0 and xysec if only and if xsecy 1

,

22 ,0:R

,11- ,:D

y = sec-1 x

o

2

3

1

1

x

y

2

2

y = sec x

,11- ,:R

,22

,0:D

x

y

2

1 1

.1x -1,x ,0y ,2

y2

and xycsc if only and if xcscy 1

2 ,00 ,

2:R

,11- ,:D

y = csc-1 x

o

2

1

1

x

y

2

y = csc x

,11- ,:R

2 ,00 ,

2:D

x

y

2

1 1

2

EXAMPLE:I. Find the exact values of the following:

2

3cosArc .1

2

2sinArc .2

2

3cosArc .3

2

3sinArc .4

2

1cosArc .5

6

.1 .Ans

4

- .2 .Ans

6

5 .3 .Ans

3

.4 .Ans

3

2 .5 .Ans

4

- .6 .Ans

6

- .7 .Ans

6

5 .8 .Ans

2

2 Sin.6 1-

2

1 Sin.7 1-

3-Cot .8 -1

2 Arcsec .9 3

.9 .Ans

2.253 Arctan .10 rad. 1.153 .10 .Ans

3

2 .11 .Ans

3

2cosCos .11 1-

3

- .12 .Ans

2

- .13 .Ans

3

2antTan .12 1-

4

-tan Sin.13 1-

2

3-Arcsin tan .14 3- .14 .Ans

0.6 .15 .Ans 0.6 tanArctan .15

17

15 .16 .Ans

15

8antcos .16 1-

9x-1 .17 .Ans 2 x3 Cos Sin.17 -1

x 2Arcsincos .18 2x-1 .18 .Ans 2

II. Solve for x:

30 x Cos - x Sin.1 0-1-1

2

3x .1 .Ans

EXERCISES:I. Find the exact values of the following:

2

2 Sin.1 1-

3Tan .2 -1

1Cot .3 -1

2

3Cos sin.4 1-

5

3Sintan .5 1-