Inverse Trigonometric Functions

We know the trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent.
But what are inverse trigonometric functions?

Inverse Trigonometric Functions: the inverse functions of the trigonometric functions with restricted domains.

Which trig functions have inverses?
-sine
-cosine
-tangent
Why?
Because these three trig functions' domains can be restricted so it can be 'one-to-one'. For a function to have an inverse, the original function must be one-to-one. This means for every value of y, there is only one x. An easy way to test if a function's one-to-one, is seeing if the function's graph passes the horizontal line test.

For Example-
We know that sine's graph looks like this:

If you were to draw a horizontal line through the graph, it intersects at more than one point. This means the function is not one-to-one. But it can be 'restricted' so that it's one-to-one.

Sine's restrictions are as follows:
Domain:
Range:

 This makes it so that the graph passes the horizontal line test, so it's one-to-one with these restrictions.

Inverse Trigonometric Functions

We know the trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent.
But what are inverse trigonometric functions?

Inverse Trigonometric Functions: the inverse functions of the trigonometric functions with restricted domains.

Which trig functions have inverses?
-sine
-cosine
-tangent
Why?
Because these three trig functions' domains can be restricted so it can be 'one-to-one'. For a function to have an inverse, the original function must be one-to-one. This means for every value of y, there is only one x. An easy way to test if a function's one-to-one, is seeing if the function's graph passes the horizontal line test.

For Example-
We know that sine's graph looks like this:

If you were to draw a horizontal line through the graph, it intersects at more than one point. This means the function is not one-to-one. But it can be 'restricted' so that it's one-to-one.

Sine's restrictions are as follows:
Domain:
Range:

This makes it so that the graph passes the horizontal line test, so it's one-to-one with these restrictions.

Cosine's restrictions:
Domain:
Range:

Tangent's restrictions:
Domain: All real numbers
Range:

For the inverse tangent graph, the (x,y) coordinates are flipped, so (x,y) becomes (y,x)
Ex: (2, 3) becomes (3, 2)

Key Inverse facts to remember:
- The output of any inverse trig function is always an angle:

-
- >These are all equivalent to each other.
-

Trigonometric Functions Review

4.1 Radian and Degree Measure

Vocab:

·      Trigonometry: “measurement of triangles”

·      Angle: determined by rotating a ray (half-line) about its endpoint

·      Initial side: starting point of the ray

·      Terminal side: position after rotation

·      Vertex: the endpoint of the ray

·      Standard position: vertex located at origin, one ray is on the positive x-axis

·      Positive angles: generated by counterclockwise rotation

·      Negative angles: clockwise rotation

·      Coterminal: angles that have common terminal side

·      Radian: measure of a central angle Ɵ that intercepts an arc s equal in lenghth to radius of circle

 

·      Complementary angles: 2 angles that add up to 90 degrees

·      Supplementary angles: 2 angles that add up to 180 degrees

* 40+140=180*

Degrees to Radians:	Radians to Degrees:

·      Arc length: Ɵ= s / r

Movie Help:

http://www.youtube.com/watch?v=7xEnSrtlkEE

Online Quiz: http://www.sporcle.com/games/akpgunner/degree_radian_conversion

4.2 Trigonometric Functions: The Unit Circle

Vocab:

·      Unit circle:

Definitions of Trigonometric Functions:

Even and Odd Trigonometric Functions

The cosine and secant functions are even.

Cos (-t)= cos t                    sec (-t) = sec t

The sine, cosecant, tangent, and cotangent functions are odd.

Sin (-t) = -sin t                    csc (-t)= -csc t

Tan (-t) = -tan t                      cot (-t)= -cot t

Movie Help:

http://www.khanacademy.org/math/trigonometry/v/unit-circle-definition-of-trig-functions

4.3 Right Triangle Trigonometry

     Sines, Cosines, and Tangents of Special Angles

Reciporcal and Phythagorean Identities:

Quotient Identities:

Online Quiz:

http://www.purposegames.com/game/937ea535

4.4 Trigonometric Functions of Any Angle

·      Reference angles: the acute angle Ɵ formed by the terminal side of Ɵ and the horizontal axis.  

Movie Help:

http://www.youtube.com/watch?v=Cg70E506maw

Online Quiz:

http://www.mathwarehouse.com/trigonometry/reference-angle/finding-reference-angle.php

4.5 Graphs of Sine and Cosine Functions

·      Parent functions:

y= sin x, y= cos x

amplitude: half the distance between the maximum and minimum values of the function

amplitude= absolute value of a

period: let b be a positive real number.

Movie Help:

http://www.youtube.com/watch?v=8nJjVku-q68

Online Help:

http://www.cliffsnotes.com/study_guide/Graphs-Sine-and-Cosine.topicArticleId-11658,articleId-11595.html

4.6 Graphs of Other Trigonometric Functions

Movie Help:

http://www.youtube.com/watch?v=jpy11Lzi1KQ

Online Help:

http://www.onlinemathlearning.com/reciprocal-function.html

4.8 Applications and Models

Example:

A bug is standing x feet away from the house. Find out how many feet the bug is away from the house?

Online Help:

http://www.algebralab.org/lessons/lesson.aspx?file=Trigonometry_TrigApplications.xml

Movie Help:

http://www.youtube.com/watch?v=-QOEcnuGQwo

Chapter 1 Review

1.1 Functions

Testing for functions

• The input value x is the number of representatives from a state and the output value y is the number of senators

Does the following equation represent y as a function of x?

                     

                     

 To each value of x there corresponds exactly one value of y.  So, y is a function of x

Evaluating a Function

Let     and find G(x+2)

 

 

 

Domain of a Function

• The domain of a function is the set of all values of the indpendent variable for which the function is defined
• If x is in the domain of   f, f   is said to be defined at x 
• If x is not in the domain of   f, f   is said to be undefined at x 
• The domain of a function can be described explicity or it can be implied by the expression used to define the function
• The implied domain is the set of all real numbers for which the expressions is defined

For instance:      

•  Another common type of implied domain is that used to avoid even roots of negative numbers

For instance:      

Evaluating a Difference Quotient

For   , find  

  
 




1.2 Graphs of Functions 

 
Max: 1
Min: 0.11 

                    Even                                                Odd


Even and odd functions
This function is odd because:









This function is even because:




1.3 Shifting, reflecting, and stretching graphs




= the -1 means it shifts horizontally one unit to the right
= the -1 means it shifts vertically one unit down

1.4 Combinations of functions 
Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to create new functions. 

Sum:   
                               
Difference:   
                                       
Product: 
                                 
Quotient:     


Identifying a composite function

Express the function as a composition of two function
                                       
=
=

1.5 InverseFunctions 

Finding inverse functions: 

Find the inverse of :   
                              
Verify that both:    and are equal to the identity function




Finding inverse functions algebraically

Find the inverse of: 


 
                                 Replace f(x) by y

                                  Interchange x and y
                                 Multiply each 
                               Isolate the y-term
                                Solve for y
                    Replace y by