Graphs of Tangent, Cotangent, Secant, and Cosecant Functions

The trigonometric functions of tan x, cot x, sec x, and csc x can be rewritten as ratios of cos x, sin x, and 1. The functions vertical asymptotes and x-intercepts can be determined with the ratios they represent. The x-intercepts are the x values that make the numerator = 0. The vertical asymptotes are the x values that make the denominator = 0. There x-intercepts and vertical asymptotes repeat every period.      

Tangent
The function f(x) = tan x can be rewritten as f(x) = sin x/cos x. The function of tan x has vertical asymptotes at the inputs that make cos x = 0 and x-intercepts at the inputs sin x = 0. The function has symmetry along the origin making it odd.
Domain: 
Range:
Vertical asymptotes: 
Period: 
X-intrecepts: 
Y-intercept: 0

http://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2d/2d_2content_6.html

Cotangent
The function f(x) = cot x can be rewritten as f(x) = cos x/sin x. The function of cot x has vertical asymptotes at the inputs that make sin x = 0 and x-intercepts at the inputs that make cos x = 0. The function has symmetry along the origin making it odd.
Domain:
Range: 
Vertical Asymptotes:
Period:
X-intercepts:
Y-intercept: none

http://www.intmath.com/trigonometric-graphs/4-graphs-tangent-cotangent-secant-cosecant.php

Secant 
The function f(x) = sec x can be rewritten as f(x) = 1/ cos x. The function of Sec x does not cross the x axis having a constant numerator of 1. It has vertical asymptotes at the inputs that make cos x = 0. Like the function of cos x the function of sec x has symmetry along the y-axis making it an even function.
Domain: 
Range:
Vertical asymptote:
Period:
X-intercepts: none
Y-intercept: o

http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Umberger/EMAT6690smu/Day6/Day6.html

Cosecant
The function f(x) = csc x = 1/ sin x Like f(x) = sec x, f(x) = csc x does not cross the x axis having a constant numerator of 1. It has a vertical asymptotes the inputs that make sin x = 0. The function has symmetry along the origin making it odd.
Domain:
Range:
Vertical asymptote:
Period: 
X-intercepts: none
Y-intercept: none

http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Umberger/EMAT6690smu/Day6/Day6.html

Inverse Trigonometric Functions

Inverse Trigonometric Functions

Inverse Sine Function

The graph of y = sin x does not pass the vertical line test due to the different values of x that have the same y value.  However, if you restrict the doman to the interval -π/2 ≤ x ≤ π/2, it becomes a one-to-one function. 

Now the inverse is possible to graph.  

The domain of y = sin x (-π/2, π/2) becomes the inverse range and the range of y = sin x (-1, 1) becomes the inverse domain.  This is called y = arcsin x.

Inverse Cosine Function

The graph of y = cos x does not pass the vertical line test due to the different values of x that have the same y value.  However, if you restrict the doman to the interval 0 ≤ x ≤ π, it becomes a one-to-one function. 

Now the inverse is possible to graph.

The domain of y = cos x (0, π) becomes the inverse range and the range of y = cos x, (-1, 1), becomes the inverse domain.  This is called y = arccos x.

Inverse Tangent Function

The tangent graph is repetitive and therefore, the domain can be restricted to the interval (-π/2, π/2).  

The domain of y = tan x (-π/2, π/2) becomes the inverse range and the range of y = tan x (-∞, ∞) becomes the inverse domain.

Notice that the vertical asymptotes of the y = tan x graph become the horizontal asymptotes of the y = arctan x graph.

Sine and Cosine Graphs

The graphs of sine and cosine functions represent their outputs for all real number values of a given angle, making the domain equal to all real numbers. The parent functions appear as shown below:

 f(x) = sin x

f(x) = cos x

The full equations of the functions can be expressed as

and

respectively, where the absolute value of a determines the amplitude of the function, determines the period, c determines the phase shift, and d determines the mid-line shift.


Definitions: 
Amplitude: half the distance between the minimum and maximum values of the function. Variations in amplitude vertically stretch and compress the graph.
Period: one completed cycle of the repeating function. Increasing the value of b thus increases the frequency of periods in a given distance.
Phase Shift: Horizontal stretching and compressing.
Mid-line Shift: Vertical translation of the graph.


Cool things to note:
- The graph of f(x)=sin x is odd, as evidenced by the reflection about the origin.
- The graph of f(x)=cos x is even, which is seen in the symmetry about the x-axis.
- If one were to shift the sine graph   units to the left, this phase shift represents the co-function identity:

Examples:

Graph:

***It is important to realize that the period is and that the phase shift is 1/2. This becomes more evident once 1/2 is factored out of the parentheses. Otherwise, the amplitude is 2/3. Together, this determines the following graph (red) compared to the parent graph (blue):

Trigonometric Functions of Any Angle

The definitions of the trigonometric functions can be expanded to incorporate (or measure) any angle. Instead of adhering to the angular constraints of a right triangle, where an angle θ must be acute so that 0° < θ < 90°, these functions can be reapplied to any other angle θ that sits on a coordinate plane in standard position with the point (x,y) on its terminal side.

 

The length of the terminal side r can be found by using the following equation derived from the Pythagorean Theorem,

 

using x and y from the coordinate point as substitutes.

The definitions of the trigonometric functions of any angle are as follows:

 

 

In order to determine the sign of the trigonometric functions, one must determine what quadrant they are in.

For example, since sin θ = y/r, that means that sin θ is positive whenever y > 0, which is in quadrants I and II (r is always positive).

Reference Angles

Suppose θ is an angle in standard position. Its reference angle is the acute angle θ' (read as "theta prime") formed by the terminal side of  angle θ and the x-axis (horizontal axis). Reference angles are calculated based on what quadrant the terminal side of θ lies in.

 

Quadrant I:

 (radians and degrees)

 

 

Quadrant II:

 (radians)

 (degrees)

 

 

Quadrant III:

 (radians)

 (degrees)

 

 

Quadrant IV:

 (radians)

 (degrees)

 

Reference angles can also be used to determine the value of a trigonometric function. For instance, suppose one must find cos 480°. One can subtract 360° from 480° to get a co-terminal angle of 120°. The reference angle of 120° (which, in this case, is θ) is 60° because 120° is in Quadrant II, so to find θ' one simply subtracts 120° from 180° to get 60°. From there, one can deduce that cos 60° is negative because it is in Quadrant II (where x < 0, given that cos θ = x/r). Therefore, cos 480° can be rewritten as -cos 60° which is -1/2.

 



In other words, to evaluate a trigonometric function of any angle, one must calculate the reference angle and determine the function value of it. From there, one must figure out the appropriate sign of the function value by looking at what quadrant it is in.