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.
Pre-Calculus A 2nd Hour, Fall 2012
Tuesday, November 20, 2012
Sunday, November 18, 2012
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.
-
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:
|
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:
4.4 Trigonometric Functions of Any Angle
Movie Help:
http://www.youtube.com/watch?v=Cg70E506maw
Online
Quiz:
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:
Online
Help:
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:
Movie Help:
http://www.youtube.com/watch?v=-QOEcnuGQwo
Saturday, November 17, 2012
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
- Another common type of implied domain is that used to avoid even roots of negative numbers
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
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