6.8 – trig inverses and their graphs...π/3 π/2 2π/3 π why are we using these values? table of...
TRANSCRIPT
![Page 1: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/1.jpg)
6.8 – TRIG INVERSES AND
THEIR GRAPHS
![Page 2: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/2.jpg)
Quick Review
How do you find inverses of functions?
Are inverses of functions always functions?
How did we test for this?
![Page 3: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/3.jpg)
Inverse Trig Functions
Original Function Inverse
y = sin x y = sin-1 x y = arcsin x
y = cos x y = cos-1 x y = arccos x
y = tan x y = tan-1 x y = arctan x
![Page 4: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/4.jpg)
I. Graphs of Inverse Trig Functions
A. Consider the graph of y = sin x
What is the domain and range of sin x?
What would the graph of y = arcsin x look like?
What is the domain and range of arcsin x?
Domain: all real numbers
Range: [-1, 1]
Domain: [-1, 1]
Range: all real numbers
![Page 5: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/5.jpg)
B. Now let’s look at y = cos x
What is the domain and range of cos x?
What would the graph of y = arccos x look like?
What is the domain and range of arccos x?
Domain: all real numbers
Range: [-1, 1]
Domain: [-1, 1]
Range: all real numbers
![Page 6: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/6.jpg)
C. Now let’s look at y = tan x
What is the domain and range of tan x?
What would the graph of y = arctanx look like?
What is the domain and range of arctan x?
![Page 7: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/7.jpg)
D. Are the inverses of sin x, cos x,
and tan x functions?
However, we can make them
functions by restricting their
domains.
Capital letters are used to distinguish when the
function’s domain is restricted.
Original Functions with
Restricted Domain
Inverse Function
y = Sin x y = Sin-1 x y = Arcsin x
y = Cos x y = Cos-1 x y = Arccos x
y = Tan x y = Tan-1 x y = Arctan x
![Page 8: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/8.jpg)
E. Original Domains Restricted
Domains
Domain Range
y = sin x
all real numbers
y = Sin x y = sin x y = Sin x
y = cos x
all real numbers
y = Cos x y = cos x y = Cos x
y = tan x
all real numbers
except n,
where n is an odd
integer
y = Tan x y = tan x
all real numbers
y = Tan x
all real numbers
![Page 9: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/9.jpg)
F. Complete the following table on your
own
Function Domain Range
y = Sin x
y = Arcsin x
y = Cos x
y = Arccos x
y = Tan xall real numbers
y = Arctan x
![Page 10: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/10.jpg)
II. Graphing with Restricted Domains
A. Table of Values of Sin x and Arcsin x
y = Sin x
X Y
-π/2
-π/6
0
π/6
π/2
y = Arcsin x
X Y
-π/2
-π/6
0
π/6
π/2
Why are we using these values?
![Page 11: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/11.jpg)
y = Sin x
X Y
-π/2 -1
-π/6 -0.5
0 0
π/6 0.5
π/2 1
y = Arcsin x
X Y
-1 -π/2
-0.5 -π/6
0 0
0.5 π/6
1 π/2
Why are we using these values?
II. Graphing with Restricted Domains
A. Table of Values of Sin x and Arcsin x
![Page 12: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/12.jpg)
Graphs of Sin x and Arcsin x
![Page 13: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/13.jpg)
B.Table of Values of Cos x and Arccos x
y = Cos x
X Y
0
π/3
π/2
2π/3
π
y = Arccos x
X Y
0
π/3
π/2
2π/3
π
Why are we using these values?
![Page 14: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/14.jpg)
Table of Values of Cos x and Arccos x
y = Cos x
X Y
0 1
π/3 0.5
π/2 0
2π/3 -0.5
π -1
y = Arccos x
X Y
1 0
0.5 π/3
0 π/2
-0.5 2π/3
-1 π
Why are we using these values?
![Page 15: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/15.jpg)
Graphs of Cos x and Arccos x
![Page 16: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/16.jpg)
C. Table of Values of Tan x and Arctan x
y = Tan x
X Y
-π/2
-π/4
0
π/4
π/2
y = Arctan x
X Y
-π/2
-π/4
0
π/4
π/2
Why are we using these values?
![Page 17: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/17.jpg)
Table of Values of Tan x and Arctan x
y = Tan x
X Y
-π/2 (veritical
asymptote)
-π/4 -1
0 0
π/4 1
π/2 (vertical
asymptote)
y = Arctan x
X Y
(horz asymptote) -π/2
-1 -π/4
0 0
1 π/4
(horizontal
asymptote)
π/2
Why are we using these values?
![Page 18: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/18.jpg)
Graphs of Tan x and Arctan x
![Page 19: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/19.jpg)
III. Writing and graphing Inverse Trig Functions
Ex 1. Write an equation for the inverse of
y = Arctan ½x. Then graph the function and its
inverse.
To write the equation:
1. Exchange x and y
2. Solve for y
x = Arctan ½y
Tan x = ½y
2Tan x = y
![Page 20: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/20.jpg)
Let’s graph 2Tan x = y first.
Complete the table:
Then graph!
y = 2Tan x
X Y
-π/2
-π/4
0
π/4
π/2
Now graph the
original function,
y = Arctan ½x by
switching the table
you just completed!
y = Arctan ½ x
X Y
Ex 1. Write an equation for the inverse of
y = Arctan ½x. Then graph the function and its inverse.
![Page 21: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/21.jpg)
Ex 2: Write an equation for the inverse of y = Sin(2x).
Then graph the function and its inverse.
To write the equation:
1. Exchange x and y
2. Solve for y
x = Sin(2y)
Arcsin(x) = 2yArcsin(x)/2 = y
![Page 22: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/22.jpg)
Write an equation for the inverse of y = Sin(2x).
Then graph the function and its inverse.
Let’s graph y = Sin(2x) first.
Why are these x-values used?
Now graph the inverse
function, y = Arcsin(x)/2 by
switching the table you just
completed!
y = Sin2x
X Y
-π/4
-π/12
0
π/12
π/4
y = Sin2x
X Y
![Page 23: 6.8 – Trig Inverses and their graphs...π/3 π/2 2π/3 π Why are we using these values? Table of Values of Cos x and Arccos x y = Cos x X Y 0 1 π/3 0.5](https://reader033.vdocuments.net/reader033/viewer/2022061002/60b112bbf36c33359b762072/html5/thumbnails/23.jpg)
IV. Evaluate each expression
See hand-written notes