Transcript 4 7 Inverse Trig

Objectives:
1. To define, graph, and
use the inverses of
sine, cosine, and
tangent
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Assignment:
P. 349: 1-16 S
P. 349: 19-34 S
P. 349: 37-42 S
P. 349: 43-48 S
P. 350: 49-58 S
P. 350: 59-64 S
P. 350-2: 71, 96, 105
You will be able to define, graph,
and use the inverse of sine, cosine,
and tangent
What does an inverse do to a function
algebraically and graphically?
Inverses switch inputs and outputs
Inverses reflect a graph over y = x
Inverses give you a way to find the input when you know the output.
Explain what x and y represent in y = sin x, then
explain what is meant by the inverse of
y = sin x.
Use the graph of y = sin x to explain why its
inverse is not a function.
The inverse of a function f is also a function iff
no horizontal line intersects the graph of f
more than once.
What could we do to the graph below so that its
inverse is cleverly a function?
We can restrict the
domain of the function
to an interval that
would pass the
horizontal line test.
In order for the inverse of y = sin x to be a
function, we have restrict the domain. Refer
again to the graph of f (x) = sin x and find a
sensible interval for which f −1(x) is a function.
While any number of intervals would work, the
most convenient is [−π/2, π/2].
Increasing, has a full range of values, contains all the
acute angles of a triangle, 1-1
The inverse sine function is defined by
y  arcsin x if and only if sin y  x
where  1  x  1 and 

2
 y

2
The inverse sine function is defined by
y  arcsin x
Read “the arcsine of x”
Means “the arc whose sine is x”
The inverse sine function is defined by
1
y  sin x if and only if sin y  x


where  1  x  1 and 
2
 y
2
The inverse sine function is defined by
1
y  sin x
Read “the inverse sine of x”
Means “the angle whose sine is x”
Does not mean reciprocal
The inverse sine function is defined by
1
y  sin x
Domain:  1,1
y  arcsin x
  
Range:   , 
 2 2
Sine ratio
Angle measure
Find the exact value of each of the following:
1
1. arcsin  
Note that there are
2
2.

3
sin  

 2 
1
3. arcsin 1
4. sin 1  3
infinitely many values of
x for which sin x = ½ , but
there is only one in the
interval [−π/2, π/2].
Graph y = arcsin x over
its entire domain!
To do this, find the
values of y = sin x
and exchange x and
y.
Refer to the graph of f (x) = cos x and find a
sensible interval for which f −1(x) is a function.
The inverse cosine function is defined by
y  arccos x if and only if cos y  x
where  1  x  1 and 0  y  
Refer to the graph of f (x) = tan x and find a
sensible interval for which f −1(x) is a function.
The inverse tangent function is defined by
y  arctan x if and only if tan y  x


where    x   and 
2
 y
2
Find the exact value of each of the following:
1
1. arccos  
2

3
2. cos   
 2 
1
3. arctan  1
4. tan 1 
3

Use a calculator to approximate each of the
following.
1. arctan (4.84)
2. arccos (−.349)
3. sin−1 (1.1)
4. arcsin (.321)
If a relation and its inverse are both functions,
then they are called inverse functions.
f  g ( x)   x and g  f ( x)   x
However, the composition of a trig function
and its inverse does not always give you x!
You have to make sure the domains and
ranges match up properly.
Interval Property
sin  arcsin x   x cos  arccos x   x tan  arctan x   x
Interval Property
arcsin  sin y   y arccos  cos y   y arctan  tan y   y
1  x  1


2
 y

2
1  x  1
0 y 
   x  


2
 y

2
If possible, find the exact value.
1. tan arctan  14 
2. sin  arcsin  
3. cos arccos .54 
4.
5 

arcsin  sin

3


To make these compositions a bit more fun, what if we
mismatched the trig functions with their inverses?
To do these problems, draw a right triangle and use
the Pythagorean Theorem.

13
 5 
csc arctan      
5
 12  

tan  
5
12
tan  
5
Quadrant II
12
Quadrant IV
Find the exact value of each of the following.

 3 
1. cos arctan   

2.
 4 

 2 
sin arccos   
 3 

When you take Calculus and you finally learn to
integrate, you’ll sometimes have to turn an
inverse trig expression into an algebraic one.
2

 x 
9

x
tan arccos    
 3 

x
cos  
x
3
9  x2
Quadrant I
Write each of the following as an algebraic
expression in x.
1. sec  arctan x 
2.
tan  arccos 2x 
Objectives:
1. To define, graph, and
use the inverses of
sine, cosine, and
tangent
•
•
•
•
•
•
•
Assignment:
P. 349: 1-16 S
P. 349: 19-34 S
P. 349: 37-42 S
P. 349: 43-48 S
P. 350: 49-58 S
P. 350: 59-64 S
P. 350-2: 71, 96, 105