# Periodic Functions

## Graphs of the Other Trigonometric Functions

We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a fire truck parked next to a warehouse. The rotating light from the truck would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and other trigonometric functions.

### Analyzing the Graph of y = tan x

We will begin with the graph of the tangent function, plotting points as we did for the sine and cosine functions. Recall that
The period of the tangent function is 
  because the graph repeats itself on intervals of 
  where 
  is a constant. If we graph the tangent function on 
  to 
  we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.
We can determine whether tangent is an odd or even function by using the definition of tangent.
Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in .
These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when 
  we can use a table to look for a trend. Because 
  and 
  we will evaluate 
  at radian measures 
  as shown in .
As 
  approaches 
  the outputs of the function get larger and larger. Because 
  is an odd function, we see the corresponding table of negative values in .
We can see that, as 
  approaches 
  the outputs get smaller and smaller. Remember that there are some values of 
  for which 
  For example, 
  and 
  At these values, the tangent function is undefined, so the graph of 
  has discontinuities at 
  At these values, the graph of the tangent has vertical asymptotes.  represents the graph of 
  The tangent is positive from 0 to 
  and from 
  to 
  corresponding to quadrants I and III of the unit circle.

### Graphing Variations of y = tan x

As with the sine and cosine functions, the tangent function can be described by a general equation.
We can identify horizontal and vertical stretches and compressions using values of 
  and 
  The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.
Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant 
  

### Graphing One Period of a Stretched or Compressed Tangent Function

We can use what we know about the properties of the tangent function to quickly sketch a graph of any stretched and/or compressed tangent function of the form 
  We focus on a single period of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function’s domain if we wish. Our limited domain is then the interval 
  and the graph has vertical asymptotes at 
  where 
  On 
  the graph will come up from the left asymptote at 
  cross through the origin, and continue to increase as it approaches the right asymptote at 
  To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use
because 
  

### Graphing One Period of a Shifted Tangent Function 

Now that we can graph a tangent function that is stretched or compressed, we will add a vertical and/or horizontal (or phase) shift. In this case, we add 
  and 
  to the general form of the tangent function.
The graph of a transformed tangent function is different from the basic tangent function 
  in several ways:

### Analyzing the Graphs of y = sec x and y = cscx

The secant was defined by the reciprocal identity 
  Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at 
  
  etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.
We can graph 
  by observing the graph of the cosine function because these two functions are reciprocals of one another. See . The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.
The secant graph has vertical asymptotes at each value of 
  where the cosine graph crosses the x-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.
Note that, because cosine is an even function, secant is also an even function. That is, 
  
As we did for the tangent function, we will again refer to the constant 
  as the stretching factor, not the amplitude.
Similar to the secant, the cosecant is defined by the reciprocal identity 
  Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at 
  
  etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.
We can graph 
  by observing the graph of the sine function because these two functions are reciprocals of one another. See . The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the cosecant function increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.
The cosecant graph has vertical asymptotes at each value of 
  where the sine graph crosses the x-axis; we show these in the graph below with dashed vertical lines.
Note that, since sine is an odd function, the cosecant function is also an odd function. That is, 
  
The graph of cosecant, which is shown in , is similar to the graph of secant.

### Graphing Variations of y = sec x and y= csc x

For shifted, compressed, and/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the cosecant function in the same way as for the secant and other functions.The equations become the following.

### Analyzing the Graph of y = cot x

The last trigonometric function we need to explore is cotangent. The cotangent is defined by the reciprocal identity 
    Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 
    etc. Since the output of the tangent function is all real numbers, the output of the cotangent function is also all real numbers.
We can graph 
    by observing the graph of the tangent function because these two functions are reciprocals of one another. See . Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.
The cotangent graph has vertical asymptotes at each value of 
    where 
    we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, 
    has vertical asymptotes at all values of 
    where 
    and 
    at all values of 
    where 
    has its vertical asymptotes.

### Graphing Variations of y = cot x

We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.

### Using the Graphs of Trigonometric Functions to Solve Real-World Problems

Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let’s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a fire truck and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function.

### Key Equations


### Key Concepts

1. The tangent function has period 
  	  
  	 
2.  
  	   is a tangent with vertical and/or horizontal stretch/compression and shift. See , , and .
3. The secant and cosecant are both periodic functions with a period of 
  	   
  	   gives a shifted, compressed, and/or stretched secant function graph. See  and .
4.  
  	   gives a shifted, compressed, and/or stretched cosecant function graph. See  and .
5. The cotangent function has period 
  	   and vertical asymptotes at 
  	  
  	 
6. The range of cotangent is 
  	   and the function is decreasing at each point in its range.
7. The cotangent is zero at 
  	  
  	 
8.  
  is a cotangent with vertical and/or horizontal stretch/compression and shift. See  and .
9. Real-world scenarios can be solved using graphs of trigonometric functions. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, match each trigonometric function with one of the following graphs.
For the following exercises, find the period and horizontal shift of each of the functions.
For the following exercises, evaluate the transformed functions.
For the following exercises, rewrite each expression such that the argument 
     is positive.

### Graphical

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.
For the following exercises, find and graph two periods of the periodic function with the given stretching factor, 
     period, and phase shift.
For the following exercises, find an equation for the graph of each function.

### Technology

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input 
     as 
    
   

### Real-World Applications
