As we learned previously for a function to have an inverse it has to pass the horizontal line test. However, because the Sine, Cosine, and Tangent graphs are periodic graphs they do not pass this test.

To find the inverse of these graphs you have to restrict the domain to the interval

Sine:                                        Inverse Sine:

Domain: ()                Domain: (-1,1)

Range: (-1,1)                           Range: ()

Cosine:                                    Inverse Cosine:

Domain: ()                    Domain:([-1,1])

Range: ([-1,1])                        Range: ()

Tangent:                                   Inverse Tangent:

Domain:                    Domain:     

Range:                         Range:

Terminology:

Sine----------ArcSin

Cosine----------ArcCos

Tangent----------ArcTan 

Graphing Sine and Cosine


The absolute value of a is the amplitude of the function y=a sinx
The range of the function y=a sinx is :

The amplitude of y= a sinx and y= a cosx represents half the distance between the maximum and minimum values of the function

The period of y= a sinbx and y = a cosbx is given by:                    

One period is the same as one cycle

Example: Sketch the graph of:

The amplitude is 1, because b=1/2.
The period is
=

=  4π

The graph will reach its first peak at π and intersect the x=axis at 2π. The period will be at 4π once the graph has made one full cycle.

Cosine parent graph

Sine parent graph



Graphs of Other TRIG Functions

Graph of the tangent function

Here is the graph of y=tan x. Note that there are vertical asymptotes where the denominator of tan x equals 0. So when cos x=0.

Graph of the Cotangent Function

Here is the graph of y=cot x. We now must consider where sin x equals zero because this is where you will put the vertical asymptotes.

Graph of the Reciprocal Functions (y= sec x)

First of all, here is the graph of y=cos x.

-Using reciprocal identities, we can draw y=sec x.

Graph of the reciprocal Functions (y=csc x)

The graph in purple is y=csc x. The graph in red is y=sin x.

Reference Angles

A Brief Journey Into The Magic Of Reference Angles





"If A is an angle in standard positon, its reference angle is the acute angle formed by the x axis and the terminal side of angle A."

-Dr Abdelkader Dendane, PH.D applied Mathematics



In layman's terms, it's the angle between A (or θ) and the nearest x axis. It is ALWAYS acute; if the angle θ is in the second quadrant, the reference angle is computed by subtracting θ from 180 degrees (180 - θ); third quadrant, by subtracting 180 from θ (θ-180); and fourth quadrant, by subtracting θ from 360 degrees (360 - θ). If the angle θ is located in the first quadrant, there is no math to be done as in that case angle θ is equal to its own reference angle.



Above is a basic illustration of reference angles.

Trigonometric Identities

Trigonometric Identities: are equalities that involve trigonometric functions and are true for every single value of the occurring variables.

Trigonometry

Radians
When measuring degrees it is often easier and more accurate to do so using radians as a unit.

Definition of a Radian:

If you want to convert an angle measured in degrees to radians, follow this process:

An example using an angle with a measure of 30°:

The process can also be used in reverse:

Standard Position
First, lets look at the rotation of angles.
Positive Rotation: counter-clockwise
Negative Rotation: clockwise

All angles have a terminal side and an initial side:

For an angle to be in standard position it must meet the following criteria:
1. Vertex at the origin
2. Initial side on x-axis

Trigonometric Functions
Remember SOHCAHTOA:

Special Cases
30-60-90 triangles:











45-45-90 triangles:

Unit Circle:
The Unit Circle shows different measurements of angles, and their positions on a graph with a circle centered around the center of a graph and with a radius of 1.

The Unit Circle is based off 30-60-90 and 45-45-90 triangles.

Even and Odd Funtions
All Triganometric funtions are even or odd.

Even Functions:



cosine
secant

Odd Funtions:



sine
tangent
cosecant
cotangent