Inverse Sine, Cosine and Tangent Functions - (6.7)
1. Inverse Sine:
Let fŸx sin x for " = t x t = . Then fŸx is an one-to-one function and f "1 Ÿx exists.
2
2
Definition: y sin "1 x means x sin y where " 1 t x t 1 and " = t y t = .
2
2
Note that:
a. Domain D sin "1 x "1, 1 ; Range R sin "1 x " = , =
2 2
b. Zero: x 0, x-intercept: Ÿ0, 0 c. y "intercept: y 0
1.5
1
0.5
d. Graph
-1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 x0.6 0.8
1
-0.5
-1
-1.5
y sin "1 x, x in ¡"1, 1¢
Example Find the exact value:
a. sin "1 1
2
e. sin "1 sin 7=
6
b. sin "1 Ÿ1 c. sin "1 " 1
2
f. cos sin "1 " 1
2
a. sin "1 1 = because sin = 1
6
6
2
2
"1
=
=
b. sin Ÿ1 , because sin
1
2
2
" = , because sin " = " sin
c. sin "1 " 1
4
4
2
1
d. sin "1 sin 3= sin "1
=
4
4
2
e. sin "1 sin 7=
sin "1 " 1 " =
6
2
6
"1
=
1
f. cos sin
"
cos "
cos = 4
4
2
3
g. tan sin "1 "
tan " = " tan = 2
3
3
=
4
d. sin "1 sin 3=
4
g. tan sin "1 "
" 1
2
1
2
" 3
Example Give the domain and range of the function. Sketch the graph of the function.
1
3
2
a. fŸx sin "1 Ÿ3x b. gŸx 2 sin "1 1 x
3
c. hŸx "3 sin "1 Ÿ2x a. fŸx sin "1 Ÿ3x 1
" 1 t 3x t 1, " 1 t x t 1 ,
3
3
1
1
D sin "1 Ÿ3x " ,
3 3
=
Range: R sin "1 Ÿ3x " , =
2 2
0.5
Domain:
-0.3
-0.2
0
-0.1
0.1
x 0.2
0.3
-0.5
-1
b. gŸx 2 sin "1 1 x
3
3
Domain: " 1 t 1 x t 1, " 3 t x t 3,
3
D sin "1 1 x "3, 3
2
1
3
Range: 2 " =
2
t 2 sin "1 1 x
3
"1 1
x t=
" = t 2 sin
3
R sin "1 1 x "=, =
t2 =
2
-3
-2
-1
0
1
x
2
3
-1
-2
3
-3
c. hŸx "3 sin "1 Ÿ2x Domain: " 1 t 2x t 1, " 1 t x t 1 ,
2
2
D "3 sin "1 Ÿ2x " 1 , 1
2 2
Range: "3 " = u "3 sin "1 Ÿ2x u "3 =
2
2
"1
3=
3=
"
t "3 sin Ÿ2x t
2
2
3=
R "3 sin "1 Ÿ2x "
, 3=
2
2
4
2
-0.4
-0.2
0
-2
-4
2. Inverse Cosine
Let fŸx cos x for 0 t x t =. Then fŸx is an 1-1 function and f "1 Ÿx exists.
Definition: y cos "1 x means x cos y where "1 t x t 1 and 0 t y t =.
2
0.2 x
0.4
Note that:
a. Domain: D cos "1 x "1, 1 ; Range: R cos "1 x 0, =
b. Zero: x 1, x-intercept: Ÿ1, 0 c. y "intercept: y =
2
3
2
1
d. Graph
-1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 x0.6 0.8
1
-1
-2
-3
y cos "1 x, x in ¡"1, 1¢
Example Find the exact value:
a. cos "1 1
2
b. cos "1 Ÿ1 e. cos "1 cos 7=
6
c. cos "1 " 1
2
f. cos cos "1 " 1
2
d. cos "1 cos 3=
4
g. tan cos "1 "
3
2
a. cos "1 1 = , because cos = 1 .
3
3
2
2
b. cos "1 Ÿ1 0, because cosŸ0 1.
= " = 3=
c. cos "1 " 1
2
4
4
3=
"1
"1
d. cos cos
cos " 1
3=
2
4
4
3
7=
"1
"1
e. cos cos
cos " 2
= " = 5=
6
6
6
3=
1
1
"1
f. cos cos "
cos
"
2
2
4
3
5=
"1
g. tan cos " 2
tan
" 1
6
3
Example Give the domain and range of the function. Sketch the graph of the function.
c. hŸx "3 cos "1 Ÿ2x a. fŸx cos "1 Ÿ3x b. gŸx 2 cos "1 1 x
3
a. fŸx cos "1 Ÿ3x 3
3
2
" 1 t 3x t 1, " 1 t x t 1 ,
3
3
1
1
D cos "1 Ÿ3x " ,
3 3
Range: R cos "1 Ÿ3x 0, =
1
Domain:
-0.3
-0.2
-0.1
0.1
x0.2
0.3
-1
-2
-3
y cos "1 Ÿ3x b. gŸx 2 cos "1 1 x
3
6
Domain: " 1 t 1 x t 1, " 3 t x t 3,
3
D cos "1 1 x "3, 3
4
2
3
Range: 2Ÿ0 t 2 cos "1 1 x t 2Ÿ= 3
"1 1
x t 2=
0 t 2 cos
3
R cos "1 1 x 0, 2=
-3
-2
-1
0
1
x 2
3
-2
-4
3
-6
c. hŸx "3 cos "1 Ÿ2x 8
Domain: " 1 t 2x t 1, " 1 t x t 1 ,
2
2
1
1
D "3 cos "1 Ÿ2x " ,
2 2
"1
Range: "3Ÿ0 u "3 cos Ÿ2x u "3Ÿ= " 3= t "3 cos "1 Ÿ2x t 0
R "3 cos "1 Ÿ2x "3=, 0
6
4
2
-0.4
-0.2
-2
0.2 x
-4
-6
-8
y "3 cos "1 Ÿ2x 3. Inverse tangent
Let fŸx tan x for " = x = . Then fŸx is an 1-1 function and f "1 Ÿx exists.
2
2
Definition: y tan "1 x means x tan y where ". x . and " = y = .
2
2
4
0.4
Note that:
"=, =
2 2
a. Domain: D tan "1 x Ÿ"., . ; Range: R tan "1 x b. Zero: x 0, x-intercept: Ÿ0, 0 c. y "intercept: y 0
d. Horizontal Asymptotes: y " = and y 2
=
2
1.5
1
0.5
e. Graph
-4
0
-2
2 x
4
-0.5
-1
-1.5
y tan "1 x, x in "., .
Example Find the exact value:
a. tan "1
1
3
e. tan "1 tan 7=
6
b. tan "1 Ÿ1 c. tan "1 " 1
3
f. cos tan "1 " 1
3
1
= , because tan "1 = 1
6
6
3
3
=
=
b. tan "1 Ÿ1 , because tan
1
4
4
" =
c. tan "1 " 1
6
3
d. tan "1 tan 3= tan "1 Ÿ"1 " =
4
4
7=
1
"1
"1
e. tan tan
tan "
"=
3
6
6
f. cos tan "1 " 1
cos " = cos = 3
6
6
=
1
1
"1
g. tan tan "
tan "
"
3
3
6
d. tan "1 tan 3=
4
g. tan tan "1 "
3
3
a. tan "1
3
2
Example Give the domain and range of the function. Sketch the graph of the function.
c. hŸx "3 tan "1 Ÿ2x a. fŸx tan "1 Ÿ3x b. gŸx 2 tan "1 1 x
3
a. fŸx tan "1 Ÿ3x 5
1.5
1
"., .
Domain: D tan "1 Ÿ3x 0.5
"=, =
2 2
Range: R tan "1 Ÿ3x -4
-2
0
2 x
4
-0.5
-1
-1.5
y tan "1 Ÿ3x , x in "., .
b. gŸx 2 tan "1
1
3
x
3
Domain: D 2 tan "1
1
3
x
"., .
Range: 2 " = t 2 tan "1 13 x
2
"= t 2 tan "1 13 x t =
R 2 tan "1
1
3
x
2
t2 =
2
"=, =
1
-15
-10
0
-5
5
x 10
15
-1
-2
-3
y 2 tan "1
"1
1
3
x , x in "., .
c. hŸx "3 tan Ÿ2x Domain: D "3 tan "1 Ÿ2x 4
"., .
Range: Ÿ"3 " = u "3 tan "1 Ÿ2x u Ÿ"3 =
2
2
3=
3=
"1
"
t "3 tan Ÿ2x t
2
2
3=
3=
R "3 tan "1 Ÿ2x "
,
2
2
2
-4
0
-2
2 x
4
-2
-4
y 2 tan "1
Example Find the exact value of the expression.
6
1
3
x , x in "., .
c. cos sin "1 " 1
3
b. sin tan "1 1
2
a. sec sin "1 Ÿz d. tan cos "1 " 1
4
a. sec sin "1 Ÿz Let 2 sin "1 Ÿz . Then
sinŸ2 z z ,
1
b. sin tan "1 1
2
"1 1
Let 2 tan
. Then
2
tanŸ2 1 ,
2
sec sin "1 Ÿz sin tan "1 1
2
secŸ2 sinŸ2 1
1 " z2
1
2 11
2
1
5
c. cos sin "1 " 1
3
"1
Let 2 sin " 1 . Then 2 is in the 4th quadrant and cosŸ2 u 0. Let 2 be the reference angle of
3
2. Then
cos sin "1 " 1
3
sinŸ2 1 ,
3
cosŸ2 cosŸ2 32 " 1
3
8
3
d. tan cos "1 " 1
4
"1
Let 2 cos " 1 . Then 2 is in the 2nd quadrant and tanŸ2 t 0. Let 2 be the reference angle of 2.
4
Then
cosŸ2 1 .
4
tan cos "1 " 1
4
tanŸ2 " tanŸ2 "
42 " 1
" 15
1
4. Inverse Secant, Inverse Cosecant and Inverse Cotangent
Relation of sec "1 x and cos "1 x : let y sec "1 x, then sec y x
1
1
"1 1
x
cos y x, cos y x , y cos
sec "1 x cos "1 1x
Relation of csc "1 x and sin "1 x : let y csc "1 x, then csc y x
1 x, sin y 1 , y sin "1 1
x
x
sin y
csc "1 x sin "1 1x
Relation of cot "1 x and tan "1 x : let y cot "1 x, then cot y x
1 x, tan y 1 , y tan "1 1
x
x
tan y
cot "1 x tan "1 1x
Example Find the exact value of each expression.
a. cot "1 Ÿ"1 a. cot "1 Ÿ"1 tan "1 Ÿ"1 " =
4
7
b.
csc "1 Ÿ2 c.
sec "1 " 2
3
b. csc "1 Ÿ2 sin "1 1
2
c. sec "1 " 2
3
8
=
6
cos "1
3
2
=
6