# 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

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

### Related manuals

advertisement