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

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

5

12

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

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