4.7 Inverse Trigonometric Functions

We have learned previously that in order for a function to have an inverse that is also a function, it must pass the horizontal line test. In other words, that function must be one-to-one, meaning that each point in the domain of the function must correspond to a unique point in that function's range.

When examining the graph of , or any other trigonometric function, we can see that the function is periodic, and therefore not one-to-one.

The inverse of would appear like so:

This is not a function. However, by constraining the domain of our function to , The graph of becomes becomes one-to-one, and therefore its inverse, shown below, is also a function.

This is known as the inverse sine function, and is denoted by either or .

In the same vein, an inverse cosine function stems from the parent function , but unlike the sine function, the domain of the cosine function is restrained to in order to maintain the function as one-to-one. The result is displayed below.

By constraining the function by the interval , an inverse tangent function may also be defined.

4.6 CSC, SEC, COT, and TAN Graphs

4.6 CSC, SEC, COT, and TAN Graphs 

Let's start with the graph of .  We know that  translates to   .

The graph of  looks like this:

We can see that in this case, the graph will never cross the x-axis.  This is because in order to find the

x-intercepts we need to set the numerator of the function,  , equal to zero.  We know that 1 cannot be equal to zero, so there will be no x-intercepts.  The asymptotes of the graph are , and , because the vertical asymptotes can be found by setting the denominator equal to zero, in this case . By looking at the Unit Circle we can find out where  is zero, wherever the y-coordinate (which is equal to ), is zero.  

Now let's move on to finding the graph of a  function.  is equal to .  

The graph of  looks like this: 

The same things as with the  function apply.  The x-intercepts are found by setting the numerator equal to zero, which in this case means there are no x-intercepts.  The vertical asymptotes will be found by finding out where  equals zero.  This again can be done by looking at where  is zero on the Unit Circle.  And since  correlates to the y-coordinate,  is the x-coordinate.  So wherever the x-coordinate is zero on the Unit Circle, is where the vertical asymptotes will be.  

In this case they are at , and .

By now you can probably see how this works.   and  functions work much the same.  Since  functions are the basis of  functions, let's look at  first.  

The graph of a  function looks like this: 

You can probably guess how we found the x-intercepts and the vertical asymptotes.  X-intercepts are found when the numerator is set equal to zero, since  is equal to , we can find the x-intercepts by finding out when  equals zero by looking at the unit circle.   equals zero at  and , and so those are the x-intercepts.  We do the same thing for finding the vertical asymptotes, except instead of looking at the numerator we look at where the denominator equals zero.  The denominator (  ) is zero at , and .  

The  functions go the opposite way of  functions.   functions are .  Here's a graph of a  function:

As we can see, the graph goes down, instead of up like the  does.  X-intercepts and vertical asymptotes are found exactly like those of all other functions- with the numerator and denominator.  

An important thing to recognize about  and  functions is that their period is shorter than those

of  and  functions.  The period of  and  functions is only , not .  The formula is as follows:  

All of the aforementioned functions undergo transformations.  They may be stretched horizontally, or vertically.  Or they may be compressed by a certain factor or shifted up/down a certain number.  This all depends on what numbers affect the function.

Here's the basic formula of a function:  .  'trig' in this case is a placeholder for any of the trigonometric functions, (, , , ,  and ).  

Here is how it works:

- The value of a is the amplitude of the function.  This means it tells us by how much the function is stretched or compressed vertically

- The value of b dictates the horizontal stretch/compression (remember that the effect it has becomes counter-intuative.  If the value of b is greater than one, the curve will horizontally compress.  If the value is less than one, it will horizontally stretch.)

- The value of c tells us if, and what, the horizontal shift is.  This means that your graph may be moved to the left or right by whatever value c is.  

- The value of d determines the vertical shift.  This means that your graph may be moved up or down by whatever value d is.