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In this lesson, we will discuss how to prepare graphical representations of the trigonometric ratios tangent, cotangent, secant, and cosecant. We will also define, discuss, and graph inverse trigonometric functions.
The basic trigonometric functions are:
You have been introduced to the six major trigonometric functions (sine, cosines, tangent, cotangent, secant, and cosecant) and their formulas, but it is important to be able to recognize the graphs of these functions as well. So, in this lesson, you will be introduced to the graphs of these functions. Before we move ahead, we will first discuss odd and even functions, since they can be recognized using these graphs.
An odd function is a function that always holds true for 
, and an even function is a function that always holds true for 
. Not all functions are either even or odd. Many (most, in fact) are neither. When looking at a graph, you can recognize an odd function from an even function by knowing that an even function is symmetric around the y-axis. This means that if you were to fold everything on the right side of the y-axis over the left side of the y-axis, you would have the same left side of the y-axis as you did before you folded over the right side onto the left side.
Many mathematicians believe that the tangent is the most important trigonometric function. We learned earlier in this module that 
. One important feature of the tangent function is that it has undefined function values at certain points. The sine and cosine functions are defined on all real numbers, but the tangent function is not.
We know that one way of getting an undefined point is by dividing a number over zero. On the unit circle, we know that 
at 90° and 270°, or 
and 
radians. We know that 
, so at 
there is an undefined point. If you move backward on the unit circle, 270° is at the same point as –90°, so there is also an undefined point at –1.56. So, both 
and 
are the undefined points.
The tangent graph is not symmetrical over the y-axis, so the tangent function is an odd function. Since all odd functions have the form 
, we can say that 
.
You are probably wondering what is so special about the tangent function. A line that is tangent to a function with respect to the x-axis can also be referred to as the slope. You should remember the definition and application of the slope from a previous module. The fact that the tangent is equal to the slope is logically reasonable because the tangent function is equal to the sine over the cosine, which is the y-coordinate over the x-coordinate, and the slope, as you have learned, is equal to rise over run.
Previously, we learned that the formula for the cotangent function is
.
Like the tangent function, the cotangent function has undefined function values at certain points. Considering the unit circle, you know that 
, and 
, so you have 
which is undefined.
The cotangent function is an odd function because it is not symmetric about the y-axis. So, since all odd functions have the form 
, we know that 
.
We know that the secant formula is
.
The graph of this formula, like those of tangent and cotangent, has undefined points. You should be able to figure out the exact points that are undefined.
You know from the unit circle above that at 90° and –90° (
and 
radians), 
. So the graph of the secant function shows undefined points at 
and at –1.57.
The graph is symmetric about the y-axis, so secant is an even function. Since all even functions have the form 
, we know that 
.
The formula for cosecant is
.
Since 
at 0° and 180°, you can see that the cosecant function is undefined at 0, 
(180°), and 
. It is undefined at 
because when moving backward on the unit circle, 180° is at the same point at –180°. You should have noticed that on these trigonometric functions, there may be (and most likely are) other undefined points, but remember that radians can be identified on a scale of infinity, and you can use the unit circle to determine where the undefined points are on a scaled graph.
The cosecant is not symmetric about the y-axis, so it is an odd function. This means that 
.
Now that you have learned how to identify odd functions and even functions, you should understand why 
and 
.
You may be able to guess what inverse trigonometric functions are. Inverse trigonometric functions can be written as 
, 
, and 
or arcsin
, arccos, and arctan. The most important thing to remember when dealing with inverse trigonometric functions is that 
, 
, and 
. Cosecant is the reciprocal of sine, while arcsin is the inverse of sine. Inverse trigonometric functions are all odd functions, so none of them are symmetric about the y-axis.
?The inverse sine function(arcsin) is just the inverse of the sine function. As mentioned above, it is NOT the reciprocal of the sine function. This function can be described the following way: 
when y is between 
and 
such that 
.
?The inverse cosine function (arcos) is the inverse of the cosine function. This function can be described in the following way: 
when y is between 
such that 
.
, and what does it look like?The inverse tangent function (arctan) is shown below, and 
when y is between 
and 
, such that 
.
Now that we know about all the basic trigonometric functions and inverse trigonometric functions, here are some standard points associated with those angles. Compare the trigonometric functions with the inverse trigonometric functions and try and to figure out the patterns yourself. Memorize these common points or remember the pattern, because they are useful and occur frequently.
If you have not figured out the pattern, take a look at the sine and cosine trigonometric functions together. You may notice that the pattern for sine is the square roots of 
and 
, and the pattern for cosine is the square roots of 
and 
.
The pattern for tangent is simply the sine divided by the cosine. Now, you should be able to determine the inverse trigonometric functions using this information.
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, 
, and 
or arcsin
, arccos, and arctan.
.
.
