Embed Size (px)
Citation preview
6.8 – TRIG INVERSES AND
THEIR GRAPHS
Quick Review
How do you find inverses of functions?
Are inverses of functions always functions?
How did we test for this?
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
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
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
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?
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
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
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
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?
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
Graphs of Sin x and Arcsin x
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?
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?
Graphs of Cos x and Arccos x
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?
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?
Graphs of Tan x and Arctan x
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
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.
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
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
IV. Evaluate each expression
See hand-written notes
