Graphs of Trigonometric
Functions
Trigonometric Inverses and their
Graphs
Section 6.8
Review
„
Inverses of Functions
„
„
The inverse of a function may be found by
interchanging the coordinates of the ordered
pairs of the function. Thus the domain of the
function becomes the range of its inverse, and
the range of the function becomes the domain
of its inverse.
Also the inverse of a function may not be a
function.
The inverse of the sine function is
called the arcsin
„
Notice the similarity of the graph of the
inverse of the sine functions to the graph
of y= sin x with the axes interchanged.
Cosine and Arccosine
Tangent and Arctangent
Notice that none of
the inverses of the
trigonometric
functions are
functions
Inverse notation
„
„
The inverse of trigonometric functions is written
two ways.
For example, the inverse of the sine function is
written as arcsine or
−
1
sin x
Do not get this confused with negative exponents.
−
1
sin x ≠
1
sin x
Example # 1
Write the equation for the inverse of
. Then graph the function and
1
y = Arc tan
x
2
its inverse.
Solution: Exchange the x and y and solve for y
„
1
x = Arc tan y
2
1
tan x = y
2
2 tan x = y
Finding the value of inverse
functions
You can use what you know about trigonometric
functions and their inverses to evaluate
expressions.
Find the value of tan −1 (sin π )
2
π
If y = sin , then y = 1
2
π
-1
o
tan 1 = 45 or
4
Example # 2
⎛
3⎞
⎟
Find cos ⎜⎜ Arctan 3 − Arc sin
⎟
2
⎝
⎠
3
Let α = Arctan 3 and β = Arcsin
2
3
Tanα = 3 and Sinβ =
2
α = 60 or
o
π
β = 60 or
3
cos(α − β ) = cos(0) = 1
o
π
3
Example # 3
„
„
sin − 1 (sin x )
Determine whether
= x
is true
or false for all values of x. If false, give a
counterexample. (Remember you only need to find one
counterexample to invalidate what appears to be a true
statement.
In the first quadrant it seems to hold true. Use your
calculator! So I try the second quadrant. When x= 120
degrees, it does not hold true. Try it!
So the statement is false.
HW # 49
„
„
„
Section 6-8
Pp. 410-411
#14, 16,22-30 all