User manual | Objectives: Assignment: To define, graph, and P. 349: 1-16 S

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

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



y



x 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

y





1 sin

x

if and only if sin

x

1 and



2

y y



x



2

The inverse sine function is defined by

y

 sin



1

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

y

 sin



1

x

Domain:





1,1



Sine ratio

y

 arcsin

x

Range:







 

,

2 2





Angle measure

Find the exact value of each of the following:

1.

arcsin

1

 

Note that there are infinitely many values of

2.

sin



1







2

3





x

for which sin x = ½ , but there is only one in the interval [−π/2, π/2].

3.

4.

sin



1

 

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



y



x x

1 and 0



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

 where

  

x

  and





2



y



x y





2

Find the exact value of each of the following:

1.

arccos

1

 

2.

cos



1







2

3





3.

arctan

 

4.

tan



1

 

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

.



( )





x

and



( )





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.

 



x x

1

 



y



2

y



2

 



x x

1

 



x

  

x

 

 



y

0

y



 



y





2



y





2

If possible, find the exact value.

1.







14



2.





3.

 

4.

 arcsin sin

5



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.

 csc arctan







5

12









 

13

5 tan







5

12

Quadrant IV tan





5



12

Quadrant II

Find the exact value of each of the following.

1.

 cos arctan







3

4









2.

 sin arccos

2

 





When you take Calculus and you finally learn to integrate, you’ll sometimes have to turn an inverse trig expression into an algebraic one.





 





 cos





x

3

x

2

Quadrant I



2

Write each of the following as an algebraic expression in x.

1.



2.



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