Section 7.4 Inverse Trigonometric Functions I

[Pages:6]Section 7.4 Inverse Trigonometric Functions I

Note: A calculator is helpful on some exercises. Bring one to class for this lecture.

OBJECTIVE 1: Sine Function

Understanding and Finding the Exact and Approximate Values of the Inverse

Sketch a graph of y = sin x (draw at least two cycles)

? The domain of y = sin x is __________________. ? Is the sine function 1-1? Why? Or why not?______________________

By

restrictingthe

domain

of

y

= sin x ,

- 2

x

2

, the function is now 1-1 and has an inverse function.

Sketch a graph of y = sin x , - x , plotting end points and several other points.

2

2

Interchange x's and y's from the graph above. Are the points on the graph of the inverse function to

y = sin x , - x below?

2

2

!

Definition( Inverse(Sine(Function(

The!inverse(sine(function,!denoted!as! y = sin-1 x ,!is!the!inverse!of!

y

=

sin

x

,

-

2

x

2

.!

!

The!domain!of!

y

=

sin -1

x

!is

-1

x

1

and!the!range!is!

-

2

y

2

.!

!!!!!!!!!!!!!!(Note!that!an!alternative!notation!for! sin-1 x !is arcsin x .)!

CAUTION:((Do(not(confuse(the(notation( sin-1

x

(with( (sin

x )-1

=

1 sin

x

=

csc

x

.(((

The(negative(1(is(not(an(exponent!((Thus,(

sin-1

x

1 sin

x

.(

(

Steps(for(Determining(the(Exact(Value(of( sin-1 x !

[ ] Step!1.! If!x"is!in!the!interval!

-1,1

,!then!the!value!of!

sin

-1

x

must!be!an!angle!in!the!interval!

!$

-

2

,

2

"%

.!

Step!2.! Let! sin-1 x = !such!that! sin = x .!

Step!3.! If! sin = 0 ,!then! = 0 !and!the!terminal!side!of!angle! !lies!on!the!positive!x#axis.!(

If! sin

>

0

,!then!

0

<

2

!and!the!terminal!side!of!angle!

!

!

!!lies!in!Quadrant!I!or!on!the!positive!y#axis.!

!

!

!

If! sin

<

0

,!then!

-

2

<

0

and!the!terminal!side!of!angle

!

!

!lies!in!Quadrant!IV!or!on!the!negative!y#axis.!!

!

!

!

!

Step!4.! Use!your!knowledge!of!the!two!special!right!triangles!and!the!graphs!of!the!trigonometric!functions,!!to!!! !!!!!!!!!!!!!!!determine!the!angle!in!the!correct!quadrant!whose!sine!is!x.!!!!!!!!!!!!!!!!!!!!!!!

" 7.4.2 Determine the exact value of the expression sin-1 $

3

% '

.

#2&

OBJECTIVE 2: Cosine Function

Understanding and Finding the Exact and Approximate Values of the Inverse

Sketch a graph of y = cos x (draw at least two cycles)

? The domain of y = cos x is __________________. ? Is the cosine function 1-1? Why? Or why not?______________________

By restricting the domain of y = cos x , 0 x , the function is now 1-1 and has an inverse function.

Sketch a graph of y = cos x , 0 x , plotting end points and several other points.

Interchange x's and y's from the graph above. Are the points on the graph of the inverse function to y = cos x , 0 x , below?

Definition( Inverse(Cosine(Function(

The!inverse(cosine(function,!denoted!as! y = cos-1 x ,!! is!the!inverse!of y = cos x , 0 x .!!

!

The!domain!of! y = cos-1 x !is -1 x 1and!the!range!is!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 0 y .(

(Note!that!an!alternative!notation!for! cos-1 x !is arccos x .)!

! !

!

!!!!!!!!!!!!!!! !

Steps(for(Determining(the(Exact(Value(of( cos-1 x (

Step(1.! If!x"is!in!the!interval! [-1,1] ,!then!the!value!of! cos-1 x must!be!an!angle!in!the!interval! [0, ].!

Step(2.! Let! cos-1 x = !such!that! cos = x .!

! ! ! ! !

!

Step(3.!

If!! cos

=

0

,!then!

=

2

!and!the!terminal!side!of!!angle

!lies!on!the!positive!y#axis.!

If! cos

>

0

,!then! 0

<

2

!and!the!terminal!side!of!angle!

!

lies!in!Quadrant!I!!or!on!the!positive!x#axis.!!

!

!

If! cos

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