inverse trig functions · remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x....
TRANSCRIPT
![Page 1: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/1.jpg)
Inverse trig functions
11/21/2011
![Page 2: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/2.jpg)
Remember: f �1(x) is the inverse function of f (x) if
y = f (x) implies f
�1(y) = x .
For inverse functions to the trigonometric functions, there are twonotations:
f (x) f
�1(x)
sin(x) sin�1(x) = arcsin(x)cos(x) cos�1(x) = arccos(x)tan(x) tan�1(x) = arctan(x)sec(x) sec�1(x) = arcsec(x)csc(x) csc�1(x) = arccsc(x)cot(x) cot�1(x) = arccot(x)
![Page 3: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/3.jpg)
In general:arc ( - ) takes in a ratio and spits out an angle:
θ!
"
#
cos(✓) = a/c so arccos(a/c) = ✓
sin(✓) = b/c so arcsin(b/c) = ✓
tan(✓) = b/a so arctan(b/a) = ✓
![Page 4: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/4.jpg)
In general:arc ( - ) takes in a ratio and spits out an angle:
θ!
"
#
cos(✓) = a/c so arccos(a/c) = ✓
sin(✓) = b/c so arcsin(b/c) = ✓
tan(✓) = b/a so arctan(b/a) = ✓
![Page 5: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/5.jpg)
In general:arc ( - ) takes in a ratio and spits out an angle:
θ!
"
#
cos(✓) = a/c so arccos(a/c) = ✓
sin(✓) = b/c so arcsin(b/c) = ✓
tan(✓) = b/a so arctan(b/a) = ✓
![Page 6: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/6.jpg)
In general:arc ( - ) takes in a ratio and spits out an angle:
θ!
"
#
cos(✓) = a/c so arccos(a/c) = ✓
sin(✓) = b/c so arcsin(b/c) = ✓
tan(✓) = b/a so arctan(b/a) = ✓
![Page 7: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/7.jpg)
There are lots of points we know on these functions...
Examples:
1. Since sin(⇡/2) = 1, we have arcsin(1) = ⇡/2
2. Since cos(⇡/2) = 0, we have arccos(0) = ⇡/2
3. arccos(1) =
0
4. arcsin(p2/2) =
⇡/4
5. arctan(1) =
⇡/4
![Page 8: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/8.jpg)
Domain/range
y = sin(x)
y = arcsin(x)
Domain: �1 x 1 Range: �⇡/2 y ⇡/2
![Page 9: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/9.jpg)
Domain/range
y = sin(x)
y = arcsin(x)
Domain: �1 x 1 Range: �⇡/2 y ⇡/2
![Page 10: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/10.jpg)
Domain/range
y = sin(x)y = arcsin(x)
Domain: �1 x 1
Range: �⇡/2 y ⇡/2
![Page 11: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/11.jpg)
Domain/range
y = sin(x)
y = arcsin(x)
-
-
!/2
"!/2
Domain: �1 x 1
Range: �⇡/2 y ⇡/2
![Page 12: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/12.jpg)
Domain/range
y = sin(x)
y = arcsin(x)
-
-
!/2
"!/2
Domain: �1 x 1 Range: �⇡/2 y ⇡/2
![Page 13: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/13.jpg)
Domain/range
y = cos(x)
y = arccos(x)
Domain: �1 x 1 Range: 0 y ⇡
![Page 14: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/14.jpg)
Domain/range
y = cos(x)
y = arccos(x)
Domain: �1 x 1 Range: 0 y ⇡
![Page 15: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/15.jpg)
Domain/range
y = cos(x)y = arccos(x)
Domain: �1 x 1
Range: 0 y ⇡
![Page 16: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/16.jpg)
Domain/range
y = cos(x)
y = arccos(x)
!-
0
Domain: �1 x 1
Range: 0 y ⇡
![Page 17: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/17.jpg)
Domain/range
y = cos(x)
y = arccos(x)
!-
0
Domain: �1 x 1 Range: 0 y ⇡
![Page 18: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/18.jpg)
Domain/range
y = tan(x)
y = arctan(x)
Domain: �1 x 1 Range: �⇡/2 < y < ⇡/2
![Page 19: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/19.jpg)
Domain/range
y = tan(x)
y = arctan(x)
Domain: �1 x 1 Range: �⇡/2 < y < ⇡/2
![Page 20: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/20.jpg)
Domain/range
y = tan(x)y = arctan(x)
Domain: �1 x 1
Range: �⇡/2 < y < ⇡/2
![Page 21: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/21.jpg)
Domain/range
y = tan(x)
y = arctan(x)
-
-
!/2
"!/2
Domain: �1 x 1
Range: �⇡/2 < y < ⇡/2
![Page 22: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/22.jpg)
Domain/range
y = tan(x)
y = arctan(x)
-
-
!/2
"!/2
Domain: �1 x 1 Range: �⇡/2 < y < ⇡/2
![Page 23: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/23.jpg)
Domain/range
y = sec(x)
y = arcsec(x)
Domain: x �1 and 1 x Range: 0 y ⇡
![Page 24: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/24.jpg)
Domain/range
y = sec(x)
y = arcsec(x)
Domain: x �1 and 1 x Range: 0 y ⇡
![Page 25: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/25.jpg)
Domain/range
y = sec(x)y = arcsec(x)
Domain: x �1 and 1 x
Range: 0 y ⇡
![Page 26: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/26.jpg)
Domain/range
y = sec(x)
y = arcsec(x)
!-
0
Domain: x �1 and 1 x
Range: 0 y ⇡
![Page 27: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/27.jpg)
Domain/range
y = sec(x)
y = arcsec(x)
!-
0
Domain: x �1 and 1 x Range: 0 y ⇡
![Page 28: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/28.jpg)
Domain/range
y = csc(x)
y = arccsc(x)
Domain: x �1 and 1 x Range: �⇡/2 y ⇡/2
![Page 29: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/29.jpg)
Domain/range
y = csc(x)
y = arccsc(x)
Domain: x �1 and 1 x Range: �⇡/2 y ⇡/2
![Page 30: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/30.jpg)
Domain/range
y = csc(x)y = arccsc(x)
Domain: x �1 and 1 x
Range: �⇡/2 y ⇡/2
![Page 31: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/31.jpg)
Domain/range
y = csc(x)
y = arccsc(x)
-
-
!/2
"!/2
Domain: x �1 and 1 x
Range: �⇡/2 y ⇡/2
![Page 32: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/32.jpg)
Domain/range
y = csc(x)
y = arccsc(x)
-
-
!/2
"!/2
Domain: x �1 and 1 x Range: �⇡/2 y ⇡/2
![Page 33: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/33.jpg)
Domain/range
y = cot(x)
y = arccot(x)
Domain: �1 x 1 Range: 0 < y < ⇡
![Page 34: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/34.jpg)
Domain/range
y = cot(x)
y = arccot(x)
Domain: �1 x 1 Range: 0 < y < ⇡
![Page 35: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/35.jpg)
Domain/range
y = cot(x)y = arccot(x)
Domain: �1 x 1
Range: 0 < y < ⇡
![Page 36: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/36.jpg)
Domain/range
y = cot(x)
y = arccot(x)
!-
0
Domain: �1 x 1
Range: 0 < y < ⇡
![Page 37: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/37.jpg)
Domain/range
y = cot(x)
y = arccot(x)
!-
0
Domain: �1 x 1 Range: 0 < y < ⇡
![Page 38: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/38.jpg)
Graphs
arcsin(x) arccos(x) arctan(x)
-
-
!/2
"!/2
!-
0
-
-
!/2
"!/2
arcsec(x) arccsc(x) arccot(x)
!-
0
-
-
!/2
"!/2
!-
0
![Page 39: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/39.jpg)
Derivatives
Use implicit di↵erentiation (just like ln(x)).
Q. Let y = arcsin(x). What is dy
dx
?
If y = arcsin(x) then x = sin(y).
Take d
dx
of both sides of x = sin(y):
LHS:d
dx
x = 1 RHS:d
dx
sin(y) = cos(y)dy
dx
= cos(arcsin(x))dy
dx
Sody
dx
=1
cos(arcsin(x)).
![Page 40: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/40.jpg)
Derivatives
Use implicit di↵erentiation (just like ln(x)).
Q. Let y = arcsin(x). What is dy
dx
?
If y = arcsin(x) then x = sin(y).
Take d
dx
of both sides of x = sin(y):
LHS:d
dx
x = 1 RHS:d
dx
sin(y) = cos(y)dy
dx
= cos(arcsin(x))dy
dx
Sody
dx
=1
cos(arcsin(x)).
![Page 41: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/41.jpg)
Derivatives
Use implicit di↵erentiation (just like ln(x)).
Q. Let y = arcsin(x). What is dy
dx
?
If y = arcsin(x) then x = sin(y).
Take d
dx
of both sides of x = sin(y):
LHS:d
dx
x = 1 RHS:d
dx
sin(y) = cos(y)dy
dx
= cos(arcsin(x))dy
dx
Sody
dx
=1
cos(arcsin(x)).
![Page 42: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/42.jpg)
Derivatives
Use implicit di↵erentiation (just like ln(x)).
Q. Let y = arcsin(x). What is dy
dx
?
If y = arcsin(x) then x = sin(y).
Take d
dx
of both sides of x = sin(y):
LHS:d
dx
x = 1 RHS:d
dx
sin(y) = cos(y)dy
dx
= cos(arcsin(x))dy
dx
Sody
dx
=1
cos(arcsin(x)).
![Page 43: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/43.jpg)
Derivatives
Use implicit di↵erentiation (just like ln(x)).
Q. Let y = arcsin(x). What is dy
dx
?
If y = arcsin(x) then x = sin(y).
Take d
dx
of both sides of x = sin(y):
LHS:d
dx
x = 1 RHS:d
dx
sin(y) = cos(y)dy
dx
= cos(arcsin(x))dy
dx
Sody
dx
=1
cos(arcsin(x)).
![Page 44: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/44.jpg)
Derivatives
Use implicit di↵erentiation (just like ln(x)).
Q. Let y = arcsin(x). What is dy
dx
?
If y = arcsin(x) then x = sin(y).
Take d
dx
of both sides of x = sin(y):
LHS:d
dx
x = 1 RHS:d
dx
sin(y) = cos(y)dy
dx
= cos(arcsin(x))dy
dx
Sody
dx
=1
cos(arcsin(x)).
![Page 45: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/45.jpg)
Simplifying cos(arcsin(x))
Call arcsin(x) = ✓.
sin(✓) = x
θ
So cos() =p
1� x
2
Sod
dx
arcsin(x) =1
cos(arcsin(x))=
1p1� x
2
.
![Page 46: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/46.jpg)
Simplifying cos(arcsin(x))
Call arcsin(x) = ✓.
sin(✓) = x
θ
So cos() =p
1� x
2
Sod
dx
arcsin(x) =1
cos(arcsin(x))=
1p1� x
2
.
![Page 47: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/47.jpg)
Simplifying cos(arcsin(x))
Call arcsin(x) = ✓.
sin(✓) = x
θ!
!
1
So cos() =p
1� x
2
Sod
dx
arcsin(x) =1
cos(arcsin(x))=
1p1� x
2
.
![Page 48: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/48.jpg)
Simplifying cos(arcsin(x))
Call arcsin(x) = ✓.
sin(✓) = x
θ!
!
1
√1"#"$²
So cos() =p
1� x
2
Sod
dx
arcsin(x) =1
cos(arcsin(x))=
1p1� x
2
.
![Page 49: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/49.jpg)
Simplifying cos(arcsin(x))
Call arcsin(x) = ✓.
sin(✓) = x
θ!
!
1
√1"#"$²
So cos(✓) =p1� x
2/1
Sod
dx
arcsin(x) =1
cos(arcsin(x))=
1p1� x
2
.
![Page 50: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/50.jpg)
Simplifying cos(arcsin(x))
Call arcsin(x) = ✓.
sin(✓) = x
θ!
!
1
√1"#"$²
So cos( arcsin(x)) =p1� x
2
Sod
dx
arcsin(x) =1
cos(arcsin(x))=
1p1� x
2
.
![Page 51: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/51.jpg)
Simplifying cos(arcsin(x))
Call arcsin(x) = ✓.
sin(✓) = x
θ!
!
1
√1"#"$²
So cos( arcsin(x)) =p1� x
2
Sod
dx
arcsin(x) =1
cos(arcsin(x))=
1p1� x
2
.
![Page 52: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/52.jpg)
Calculate
d
dx
arctan(x).
1. Rewrite y = arctan(x) as x = tan(y).
2. Use implicit di↵erentiation and solve for dy
dx
.
3. Your answer will have sec(arctan(x)) in it.Simplify this expression using
arctan(x)
!
! 1
dy
dx
=1
(sec(arctan(x)))2=
1
1 + x
2
![Page 53: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/53.jpg)
Recall: In general, if y = f
�1(x), then x = f (y).
So 1 = f
0(y)dydx
= f
0 �f
�1(x)�, and so
d
dx
f
�1(x) =1
f
0(f (x))
f (x) f
0(x)
cos(x) � sin(x)
sec(x) sec(x) tan(x)
csc(x) � csc(x) cot(x)
cot(x) � csc2(x)
f (x) f
0(x)
arctan(x) � 1
sin(arccos(x))
arcsec(x) 1
sec(arcsec(x)) tan(arcsec(x))
arccsc(x) � 1
csc(arccsc(x)) cot(arccsc(x))
arccot(x) � 1�csc(arccot(x))
�2
![Page 54: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/54.jpg)
Recall: In general, if y = f
�1(x), then x = f (y).
So 1 = f
0(y)dydx
= f
0 �f
�1(x)�, and so
d
dx
f
�1(x) =1
f
0(f (x))
f (x) f
0(x)
cos(x) � sin(x)
sec(x) sec(x) tan(x)
csc(x) � csc(x) cot(x)
cot(x) � csc2(x)
f (x) f
0(x)
arctan(x) � 1
sin(arccos(x))
arcsec(x) 1
sec(arcsec(x)) tan(arcsec(x))
arccsc(x) � 1
csc(arccsc(x)) cot(arccsc(x))
arccot(x) � 1�csc(arccot(x))
�2
![Page 55: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/55.jpg)
Recall: In general, if y = f
�1(x), then x = f (y).
So 1 = f
0(y)dydx
= f
0 �f
�1(x)�, and so
d
dx
f
�1(x) =1
f
0(f (x))
f (x) f
0(x)
cos(x) � sin(x)
sec(x) sec(x) tan(x)
csc(x) � csc(x) cot(x)
cot(x) � csc2(x)
f (x) f
0(x)
arctan(x) � 1
sin(arccos(x))
arcsec(x) 1
sec(arcsec(x)) tan(arcsec(x))
arccsc(x) � 1
csc(arccsc(x)) cot(arccsc(x))
arccot(x) � 1�csc(arccot(x))
�2
![Page 56: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/56.jpg)
To simplify, use the triangles
arccos(x)
!!
1arcsec(x)
!
! 1
arccsc(x)
!
!
1arccot(x)
!!
1
![Page 57: Inverse trig functions · Remember: f1(x) is the inverse function of f(x)if y = f(x)impliesf1(y)=x. For inverse functions to the trigonometric functions, there are two notations:](https://reader035.vdocument.in/reader035/viewer/2022070114/608840392039b658e70d6ba9/html5/thumbnails/57.jpg)
More examples
Since d
dx
arctan(x) = 1
1+x
2
, we know
1. d
dx
arctan(ln(x)) =
1
1 + (ln(x))2· 1x
2.
Z1
1 + x
2
dx =
arctan(x) + C
3.
Z1
(1 + x)px
dx =
Z1⇣
1 +�p
x
�2
⌘px
dx
= 1
2
arctan(px) + C