Graphing the Other Four Trigonometric Functions

Just like the graphs of the sin x and cos x, the graphs of the other four functions have very distinct patterns they can be recognized by. In this blog post we'll be covering the graphs of tan x, cot x, csc, and sec x

These functions also come in the formula        f(x) = a trig[b(x-c)] + d  

The tan x Graph:

To graph the function of tan x, we look back at graphing rational functions to make it easier on us. 

Since tan x equals sin x divided by cos x, we can use that instead. 

By taking each value for the angles in the unit circle and plugging them into the new function, we can find plot-able values for tan x.

When we plot those points, it creates a graph that looks like this. Some important things to remember about the parent tan x graph are:

1. x-intercept at (0,0)

2. Values that make cos x, or the denominator zero, are vertical asymptotes of the graph

3. Values that make sin x, or the numerator zero, are horizontal asymptotes of the graph.

4. The period of tan x is pi

Like sin x and cos x, the value of a vertically stretches and compresses the graph but because the graph approaches infinity when it approaches the asymptotes, you can only notice the difference a makes when compared to another tan x graph. If it is negative however, it will be reflected across the x-axis. 

The value of b affects the period of the graph. If it is negative, it will reflect the graph across the y-axis. This looks exactly like reflecting it across the y-axis however. Unlike sin x and cos x, the formula for finding 

the period or b is: 

 Finally, the values of c and d just shift the graph either to the left or right, or up and down, respectively. Remember that the x within the parentheses needs to have a coefficient of 1. Also within the parentheses, positive means left, negative means right. 

The cot x Graph:

The cot x graph works just like the tan x graph.

So by looking at the table towards the top, we just take the values for cos x and divide them by the values for sin x. This gets us the values for cot x.

Plotting the points gets us the graph:

Important things about the parent cot x graph:

1. x = 0 is a vertical asmyptote 

2. Values that make sin x, or the denominator zero, are vertical asymptotes of the graph

3. Values that make cos x, or the numerator zero, are horizontal asymptotes of the graph.

4. The period of cot x is pi

The values of a, b, c, and d all work the same way they do with the tan x graph. It also uses the formula  

The csc x Graph:

We also make csc x look more like a rational function by doing:

Looking at the table at the top of the graph, we just divide 1 by the values we see for sin x, then we plot. This gets us a graph that looks like this: 

The graphs of csc x and sec x look a little bit different than x and cot x. These graphs make parabola looking things that are boxed in by vertical asymptotes. 

Important things to note about the parent csc x graph:

1. x = 0 is a vertical asymptote 

2. The values of that make sin x equal zero are vertical asmptotes

3. The period is 2 pi

4. The average distance between the minimums and maximums is y = 0

Changing the value of a will change how far away the minimums and maximums are from the "middle line" between. To find the "middle line" of the graph, just add the y values of a minimum and maximum together and divide by 2. If it is negative, the graph reflects across the x-axis. 

Changing the value of b affects the period of the graph. Like the sin x graph, the period for the graph of csc is 2 pi. To find the period, use the formula:

Changing the the value of c shifts the graph to the left or right, but pay attention to the sign of your a and b values so you catch which way the graph is actually moves. 

Changing the value of d shifts the graph up or down. In the case of csc x, it is going to shift the "middle line" up or down. 

The sec x Graph:

Same process, turn sec x into a rational function. So, 

Looking at the table above, we find all the values for sec x and then we plot.

That gets us this graph: 

Important things to note about the parent graph of sec x:

1. x = 0 goes right down the center of the the parabola-like shapes 

2. The values of that make cos x equal zero are vertical asmptotes

3. The period is 2 pi

4. The average distance between the minimums and maximums is y = 0

And finally, the values of a, b, c, and d affect the graph in the same way as they do in the graph of csc x.

So that's nearly everything you need to know about graphing the other four trigonometric functions. Hope this helps!