math12 lesson7
TRANSCRIPT
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-