Domain and Range of Trig and Inverse Trig Functionsweim0024/pdf/15 - Domain... · Domain and Range of General Functions The domain of a function is the list of all possible inputs

Domain and Range of Trig and Inverse TrigFunctions

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Preliminaries and Objectives

Preliminaries:• Graphs of y = sin x , y = cos x and y = tan x .

Objectives:• Find the domain and range of basic trig and inverse trig

functions.

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range of General Functions

• The domain of a function is the list of all possible inputs(x-values) to the function.

• The range of a function is the list of all possible outputs(y -values) of the function.

• Graphically speaking, the domain is the portion of thex-axis on which the graph casts a shadow.

• Graphically speaking, the range is the portion of the y -axison which the graph casts a shadow.

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x)

−∞ < x <∞ − 1 ≤ y ≤ 1

y = cos(x)

−∞ < x <∞ − 1 ≤ y ≤ 1

y = tan(x)

x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x)

− 1 ≤ x ≤ 1 − π2 ≤ y ≤ π

2

y = cos−1(x)

− 1 ≤ x ≤ 1 0 ≤ y ≤ π

y = tan−1(x)

−∞ < x <∞ − π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x) −∞ < x <∞

− 1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞

− 1 ≤ y ≤ 1

y = tan(x)

x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x)

− 1 ≤ x ≤ 1 − π2 ≤ y ≤ π

2

y = cos−1(x)

− 1 ≤ x ≤ 1 0 ≤ y ≤ π

y = tan−1(x)

−∞ < x <∞ − π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞

− 1 ≤ y ≤ 1

y = tan(x)

x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x)

− 1 ≤ x ≤ 1 − π2 ≤ y ≤ π

2

y = cos−1(x)

− 1 ≤ x ≤ 1 0 ≤ y ≤ π

y = tan−1(x)

−∞ < x <∞ − π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = tan(x)

x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x)

− 1 ≤ x ≤ 1 − π2 ≤ y ≤ π

2

y = cos−1(x)

− 1 ≤ x ≤ 1 0 ≤ y ≤ π

y = tan−1(x)

−∞ < x <∞ − π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = tan(x) x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . .

−∞ < y <∞

y = sin−1(x)

− 1 ≤ x ≤ 1 − π2 ≤ y ≤ π

2

y = cos−1(x)

− 1 ≤ x ≤ 1 0 ≤ y ≤ π

y = tan−1(x)

−∞ < x <∞ − π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = tan(x) x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x)

− 1 ≤ x ≤ 1 − π2 ≤ y ≤ π

2

y = cos−1(x)

− 1 ≤ x ≤ 1 0 ≤ y ≤ π

y = tan−1(x)

−∞ < x <∞ − π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = tan(x) x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x) − 1 ≤ x ≤ 1

− π2 ≤ y ≤ π

2

y = cos−1(x) − 1 ≤ x ≤ 1

0 ≤ y ≤ π

y = tan−1(x)

−∞ < x <∞ − π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = tan(x) x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x) − 1 ≤ x ≤ 1 − π2 ≤ y ≤ π

2

y = cos−1(x) − 1 ≤ x ≤ 1

0 ≤ y ≤ π

y = tan−1(x)

−∞ < x <∞ − π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = tan(x) x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x) − 1 ≤ x ≤ 1 − π2 ≤ y ≤ π

2

y = cos−1(x) − 1 ≤ x ≤ 1 0 ≤ y ≤ π

y = tan−1(x)

−∞ < x <∞ − π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = tan(x) x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x) − 1 ≤ x ≤ 1 − π2 ≤ y ≤ π

2

y = cos−1(x) − 1 ≤ x ≤ 1 0 ≤ y ≤ π

y = tan−1(x) −∞ < x <∞

− π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain and Range

Function Domain Range

y = sin(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞ − 1 ≤ y ≤ 1

y = tan(x) x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x) − 1 ≤ x ≤ 1 − π2 ≤ y ≤ π

2

y = cos−1(x) − 1 ≤ x ≤ 1 0 ≤ y ≤ π

y = tan−1(x) −∞ < x <∞ − π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain, Range and Graphs

y = sin x y = cos x

x

y

π2

π 3π2

2π

1

−1

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain, Range and Graphs

y = tan x

x

y

−π2

π2

1

−1

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain, Range and Graphs

y = sin−1 x

x

y

1

−1

(−1,−π2 )

(1, π2 )

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain, Range and Graphs

y = cos−1 x

x

y

π2

(−1, π)

(1,0)

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Domain, Range and Graphs

y = tan−1 x

x

y

1

−1

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Recap

Function Domain Range

y = sin(x) −∞ < x <∞ −1 ≤ y ≤ 1

y = cos(x) −∞ < x <∞ −1 ≤ y ≤ 1

y = tan(x) x 6= . . .− π2 ,

π2 ,

3π2 ,

5π2 . . . −∞ < y <∞

y = sin−1(x) −1 ≤ x ≤ 1 −π2 ≤ y ≤ π

2

y = cos−1(x) −1 ≤ x ≤ 1 0 ≤ y ≤ π

y = tan−1(x) −∞ < x <∞ −π2 < y < π

2

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Credits

Written by: Mike Weimerskirch

Narration: Mike Weimerskirch

Graphic Design: Mike Weimerskirch

University of Minnesota Domain and Range of Trig and Inverse Trig Functions

Copyright Info

c© The Regents of the University of Minnesota & MikeWeimerskirchFor a license please contact http://z.umn.edu/otc

University of Minnesota Domain and Range of Trig and Inverse Trig Functions