Summary: Many of the computational requirements for an introductory physics course involve trigonometry. This module provides a brief tutorial on trigonometry fundamentals that is designed to be accessible to blind students.
This module is part of a collection of modules designed to make physics concepts accessible to blind students.
See http://cnx.org/content/col11294/latest/ for the main page of the collection and http://cnx.org/content/col11294/latest/#cnx_sidebar_column for the table of contents for the collection.
The collection is intended to supplement but not to replace the textbook in an introductory course in high school or college physics.
Many of the computational requirements for an introductory physics course involve trigonometry. This module provides a brief tutorial on trigonometry fundamentals that is designed to be accessible to blind students.
In addition to an Internet connection and a browser, you will need the following tools (as a minimum) to work through the exercises in these modules:
The minimum prerequisites for understanding the material in these modules include:
I recommend that you open another copy of this document in a separate browser window and use the following links to easily find and view the figures and listings while you are reading about them.
I recommend that you also study the other lessons in my extensive collection of online programming tutorials. You will find a consolidated index at www.DickBaldwin.com .
Many of the computational requirements for an introductory physics course involve trigonometry. This module provides a brief tutorial on trigonometry fundamentals that is designed to be accessible to blind students.
Sine, cosine, and tangent
There are many topics, such as identities, that are covered in an introductory trigonometry course that won't be covered in this module. Instead, this module will concentrate mainly on performing computations on right angles using the sine, cosine, and tangent of an angle.
If I find it necessary to deal with identities in a later module, I will come back and update this module accordingly.
Download files
You will need to download two svg graphics files to complete the work in this module. Click this link to download a zip file named Phy1020.zip containing those svg files.
If you don't already have it, you may also need to download and install the free IVEO Viewer software. As of this writing, the Viewer is available for downloading at http://www.viewplus.com/products/software/handsonlearning/ .
Graph board and protractor
Unless you can create tactile graphics on paper, you will need your graph board and your protractor to perform the exercises in this module. Please prepare your graph board with perpendicular horizontal and vertical axes with the origin located near the center of the graph board.
The most common unit of angular measurement used by the general public is the degree. As you are probably aware, there are 360 degrees in a circle.
The most common unit of angular measurement used by scientists and engineers is the radian.
(If you would like more background on radians, go to http://www.clarku.edu/~djoyce/trig/ .)
Conversions between radians and degrees
You may or may not be aware that one radian is equal to approximately 57.3 degrees. It is easier to remember, however, that 180 degrees is equal to PI radians where PI is the mathematical constant having an approximate value of 3.14159. We will use this latter relationship extensively to convert from degrees to radians and to convert from radians to degrees while working through the exercises in these modules.
An exercise involving degrees and radians
Let's do a short exercise involving degrees and radians. Please create an html file containing the code shown in Listing 1 and open it in your browser.
<! File JavaScript01.html >
<html><body>
<script language="JavaScript1.3">
function toRadians(degrees){
return degrees*Math.PI/180
}//end function toRadians
//============================================//
function toDegrees(radians){
return radians*180/Math.PI
}//end function toDegrees
//============================================//
var degrees = 90
var radians = toRadians(degrees)
document.write("degrees = " + degrees +
" : radians = " + radians + "</br>")
radians = 1
degrees = toDegrees(radians)
document.write("radians = " + radians +
" : degrees = " + degrees + "</br>")
radians = Math.PI
degrees = toDegrees(radians)
document.write("radians = " + radians +
" : degrees = " + degrees)
</script>
</body></html>
Output for script in Listing 1
When you open the file in your browser, the text shown in Figure 1 should be displayed in the browser window.
Output for script in Listing 1.  


The toRadians and toDegrees functions
Because it will frequently be necessary for us to convert between degrees and radians, I decided to write two functions that we will use to make those conversions. That will eliminate the need for us to stop and think about the conversion (and possibly get it backwards) when writing code. We will simply call the function that performs the conversion in the required direction.
The toRadians function expects to receive an input parameter describing an angle in degrees and returns the value for that same angle in radians.
The toDegrees function expects to receive an input parameter describing an angle in radians and returns the value for that same angle in degrees.
Global variables named degrees and radians
The code in Listing 1 begins by declaring global variables named degrees and radians . The variable named degrees is initialized to 90 degrees.
The toRadians function is called to convert that value of degrees to radians. The returned value in radians is stored in the variable named radians .
Display contents of both variables
Then the document.write method is called to display the values contained in both variables, producing the first line of output text shown in Figure 1 .
Modify variable contents, convert, and display again
Following that, a value of 1 is assigned to the variable named radians . The toDegrees function is called to convert that value to degrees, and the result is stored in the variable named degrees .
Once again, the document.write method is called to display the current contents of both variables, producing the second line of output text shown in Figure 1 .
One more time
Finally, the mathematical constant, PI, is stored in the variable named radians . Then that value is converted to degrees and stored in the variable named degrees . The current values in both variables are displayed, producing the last line of output text shown in Figure 1.
And the results were...
As you can see from Figure 1 ,
A template
You might want to save your html file as a template for use with future exercises that require conversions between radians and degrees. This will be particularly useful when we write scripts that use JavaScript's builtin trigonometric methods. Those methods deal with angles almost exclusively in radians while we tend to think of angles in degrees. We will use these two functions to perform conversions between degrees and radians when required.
An exercise involving a right triangle
For the next exercise, I would like for you to create a right triangle on your graph board by placing pushpins at the following coordinates:
(If you are able to use the svg file mentioned below to create tactile graphics, it probably won't be necessary for you to do the graphboard exercise. You can explore the tactile graphics instead.)
The vertices of a right triangle
Each pushpin represents a vertex of a right triangle. If you enclose all three pushpins with a rubber band, you will have "drawn" a right triangle with its base on the horizontal axis.
The base of the triangle will have a length of three units. Another side of the triangle will have a length of four units. The hypotenuse of the triangle will have still another length.
The angle at the origin
The base and the hypotenuse will form an angle at the origin opening outward to the right. As mentioned before, the base will be on the horizontal axis. The side that connects the base and the far end of the hypotenuse will be parallel to the vertical axis.
Names for the three sides of the right triangle
Lets establish some names for the three sides of the right triangle when located on the graph board in this manner.
We will continue to refer to the hypotenuse as the hypotenuse and abbreviate it hyp . We will refer to the base as the adjacent side relative to the angle at the origin and abbreviate it adj . We will refer to the third side as the opposite side relative to the angle at the origin and abbreviate it opp .
Tactile graphics  the file named Phy1020b1.svg
Much of the material in the next several paragraphs will only make sense to you if you are very familiar with the module named Manual Creation of Tactile Graphics at http://cnx.org/content/m38546/latest/ .
When you downloaded the zip file named Phy1020.zip using the link given above , you should have found that the zip file contains a file named Phy1020b1.svg.
The purpose of this file is to make it possible for you to create tactile graphics for the right triangle. The procedure for creating the tactile graphics is explained in the earlier module named Manual Creation of Tactile Graphics at http://cnx.org/content/m38546/latest/ .
Image from file Phy1020b1.svg
For the benefit of any sighted persons that may be assisting you, Figure 2 shows a reduced version of the graphic contained in the file named Phy1020b1.svg. This is a mirror image of the image that is to be presented to the student after embossing. A nonmirrorimage version is presented in Figure 19 .
Mirror image from file Phy1020b1.svg. 

Page Setup information
If you use the IVEO Viewer software to print this svg file, you will need to set up the page by selecting Page Setup on the IVEO File menu. The Page Setup information for this file is shown in Figure 3 . These are the default settings for Letter (Portrait) in IVEO.
Page Setup for file Phy1020b1.svg.  


Braille keys
Note that the file named Phy1020b1.svg and Figure 2 contain keys using the characters from "m" through "z" and "A" that can be used to create Braille characters during manual embossing. The key characters are shown in an oblique font that is smaller than the normal text. The purpose of these keys is explained in the earlier module titled Manual Creation of Tactile Graphics at http://cnx.org/content/m38546/latest/ .
Keyvalue pairs
Figure 4 contains the text values associated with each of the Braille keys.
Text values for Braille keys in file Phy1020b2svg.  


The length of the hypotenuse
Now that you have your right triangle on the graph board, or you have access to tactile graphics created from the svg file, and you know the lengths of the adjacent and opposite sides, do you remember how to calculate the length of the hypotenuse?
The Pythagorean theorem
Hopefully you know that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two other sides. Thus, the length of the hypotenuse is equal to the square root of the sum of the squares of the other two sides.
In this case we can do the arithmetic in our heads to compute the length of the hypotenuse. (I planned it that way.)
The square of the adjacent side is 9. The square of the opposite side is 16. The sum of the squares is 25, and the square root of 25 is 5. Thus, the length of the hypotenuse is 5.
A 345 triangle
You have created a rather unique triangle. You have created a right triangle in which the sides are either equal to, or proportional to the integer values 3, 4, and 5.
I chose this triangle on purpose for its simplicity. We will use it to investigate some aspects of trigonometry.
You will often hear people talk about the sine of an angle or the cosine of an angle. Just what is the sine of an angle anyway?
Although the sine of an angle is based on very specific geometric considerations involving circles (see http://www.clarku.edu/~djoyce/trig/ ), for our purposes, the sine of an angle is simply a ratio between the lengths of two different sides of a right triangle.
A ratio of two sides
For our purposes, we will say that the sine of an angle is equal to the ratio of the opposite side and the hypotenuse. Therefore, in the case of the 345 triangle that you have on your graph board, the sine of the angle at the origin is equal to 4/5 or 0.8.
If we know the lengths of the hypotenuse and the opposite side, we can compute the sine and use it to determine the value of the angle. (We will do this later using the arcsine.)
Conversely, if we know the value of the angle but don't know the lengths of the hypotenuse and/or the opposite side, we can obtain the value of the sine of the angle using a scientific calculator (such as the Google calculator) or lookup table.
The sine of an angle  sample computation
Enter the following into the Google search box:
sin(53.13010235415598 degrees)
The following will appear immediately below the search box:
sin(53.13010235415598 degrees) = 0.8
This matches the value that we computed above as the ratio of the opposite side and the hypotenuse.
The arcsine (inverse sine) of an angle
The arcsine of an angle is the value of the angle having a given sine value. In other words, if you know the value of the sine of an unknown angle, you can use a scientific calculator or lookup table to find the value of the angle.
For example, we know that the sine of the angle at the origin on your graph board is 4/5. From that, we can determine the value of the angle. However, we probably can't do this calculation in our heads so we will use the Google calculator to compute the value of the angle.
The arcsine of an angle  sample computation
Enter the following into the Google search box:
arcsin(4/5) in degrees
The following will appear immediately below the search box:
arcsin(4/5) = 53.1301024 degrees
This is the angle that corresponds to a ratio of the opposite side to the hypotenuse of 4/5.
We can also write a JavaScript script to perform the calculation, which we will do shortly.
Getting the angle for a known sine value
Please use your protractor to measure and record the angle at the origin on your graph board, or measure it on your tactile graphic. Then create an html file containing the code shown in Listing 2 and open it in your browser.
<! File JavaScript02.html >
<html><body>
<script language="JavaScript1.3">
function toRadians(degrees){
return degrees*Math.PI/180
}//end function toRadians
//============================================//
function toDegrees(radians){
return radians*180/Math.PI
}//end function toDegrees
//============================================//
var opp = 4
var hyp = 5
var ratio = opp/hyp
var angRad = Math.asin(ratio)
var angDeg = toDegrees(angRad)
document.write("radians = " + angRad + "</br>")
document.write("degrees = " + angDeg)
</script>
</body></html>
The output for the angle
When you open your html file in your browser, the output shown in Figure 5 should appear in the browser window.
Output for script in Listing 2.  


Did you measure the angle to be 53 degrees with your protractor. If so, congratulations. If not, you should probably take another look at it.
Define conversion functions
The code in Listing 2 begins by defining the functions named toRadians and toDegrees that we developed earlier in Listing 1 . (In this case, we will only need the function named toDegrees so I could have omitted the code for the function named toRadians .)
Declare and initialize variables
Then the code in Listing 2 declares and initializes variables to represent the lengths of the opposite side and the hypotenuse for the triangle on your graph board ( opp and hyp ). Then it computes and saves the ratio of the two. (We learned earlier that the ratio is the value of the sine of the angle at the origin even though we don't know the value of the angle.)
The builtin Math.asin method
JavaScript has a builtin method named Math.asin that receives the sine value for an unknown angle and returns the value of the corresponding angle in radians. (The Math.asin method has the same purpose at the word arcsin in the Google calculator.)
The returned value is an angle between PI/2 and PI/2 radians. (I will have more to say about this later.)
Listing 2 calls the Math.asin method, passing the ratio (sine of the angle) as a parameter, and stores the returned value in a variable named angRad .
Then Listing 2 calls the toDegrees method, passing the value of angRad as a parameter and stores the returned value in a variable named angDeg .
Finally, Listing 2 calls the document.write method twice in success to display the angle values shown in Figure 5 .
Another exercise with a different viewpoint
Now let's approach things from a different viewpoint. Assume that
Assume also that for some reason you can't simply measure the length of the opposite side. Therefore, you must calculate it. This is a common situation in physics, so let's see if we can write a script that will perform that calculation for us.
Please create an html file containing the code shown in Listing 3 and open the file in your browser.
<! File JavaScript03.html >
<html><body>
<script language="JavaScript1.3">
function toRadians(degrees){
return degrees*Math.PI/180
}//end function toRadians
//============================================//
function toDegrees(radians){
return radians*180/Math.PI
}//end function toDegrees
//============================================//
var hyp = 5
var angDeg = 53.13
var angRad = toRadians(angDeg)
var sine = Math.sin(angRad)
var opp = hyp * sine
document.write("opposite = " + opp + "</br>")
hyp = opp/sine
document.write("hypotenuse = " + hyp + "</br>")
</script>
</body></html>
The output for the opposite side
When you open your html file in your browser, the output shown in Figure 3 should appear in your browser window.
Output for script in Listing 3.  


Computing length of opposite side with the Google calculator
We could also compute the length of the opposite side using the Google calculator.
The length of the opposite side  sample computation
Enter the following into the Google search box:
5*sin(53.1301024 degrees)
The following will appear immediately below the search box:
5 * sin(53.1301024 degrees) = 4
This is the length of the opposite side for the given angle and the given length of the hypotenuse.
Interesting equations
We learned earlier that the sine of the angle is equal to the ratio of the opposite side and the hypotenuse. We also learned that the angle is the arcsine of that ratio.
If we know any two of those values ( angle , opp , hyp ), we can find the third (with a little algebraic manipulation) as shown in Figure 7 .
Interesting sine equations.  


Getting back to Listing 3
After defining the radian/degree conversion functions, Listing 3 declares and initializes variables representing the length of the hypotenuse and the angle in degrees. (Note that the angle in degrees was truncated to four significant digits, which may introduce a slight inaccuracy into the computations.)
Get and use the sine of the angle
That angle is converted to radians and passed as a parameter to the Math.sin method, which returns the value of the sine of the angle.
The value for the sine of the angle is then used in an algebraic equation to compute the length of the opposite side, which is displayed in Figure 6 . (This equation is one of the equations shown in Figure 7 .)
Looks very close to me
As you can see, the computed value for the opposite side shown in Figure 6 is extremely close to the known value of 4 units.
Recompute the length of the hypotenuse
After that, the value of the hypotenuse is recomputed (as though it were the unknown in the problem) using the value of the sine and the recently computed value of the opposite side. (Once again, one of the equations from Figure 7 is used to perform the computation.) The output length for the hypotenuse is shown in Figure 6 , and it matches the known value.
Example usage of Math.asin and Math.sin methods
Listing 2 and Listing 3 provide examples of how to use the JavaScript Math.asin and Math.sin methods to find the angle, the opposite side, or the hypotenuse of a right triangle when the other two are known as shown by the equations in Figure 7 .
You are going to find the discussion in this section to be very similar to the discussion in the previous section on the sine and the arcsine of an angle.
Once again, although the cosine of an angle is based on very specific geometric considerations involving circles (see http://www.clarku.edu/~djoyce/trig/ ), for our purposes, the cosine of an angle is simply a ratio between the lengths of two different sides of a right triangle.
A ratio of two sides
For our purposes, we will say that the cosine of an angle is equal to the ratio of the adjacent side and the hypotenuse. Therefore, in the case of the 345 triangle that you have on your graph board, the cosine of the angle at the origin is equal to 3/5 or 0.6.
As before, if we know the lengths of the hypotenuse and the adjacent side, we can compute the cosine and use it to determine the value of the angle. (We will do this later.)
Conversely, if we know the value of the angle but don't know the lengths of the hypotenuse and/or the adjacent side, we can obtain the cosine value (the ratio of the adjacent side and the hypotenuse) using a scientific calculator or lookup table and use it for other purposes later.
The cosine of an angle  sample computation
Enter the following into the Google search box:
cos(53.13010235415598 degrees)
The following will appear immediately below the search box:
cos(53.13010235415598 degrees) = 0.6
This matches the ratio of the adjacent side to the hypotenuse for a 345 triangle.
The arccosine (inverse cosine) of an angle
The arccosine of an angle is the value of the angle having a given cosine value. In other words, if you know the value of the cosine of an unknown angle, you can use a scientific calculator or lookup table to find the value of the angle.
Getting the angle for a known cosine value
For example, we know that the cosine of the angle at the origin on your graph board is 0.6. From that, we can determine the value of the angle using either the Google calculator or JavaScript.
The arccosine of an angle  sample computation
Enter the following into the Google search box:
arccos(3/5) in degrees
The following will appear immediately below the search box:
arccos(3/5) = 53.1301024 degrees
This is the angle that corresponds to a ratio of the adjacent side to the hypotenuse of 3/5.
As you should expect. the computed angle is the same as before. We didn't change the angle, we simply computed it using a different approach.
Getting the angle using JavaScript
Please create an html file containing the code shown in Listing 4 and open it in your browser.
<! File JavaScript04.html >
<html><body>
<script language="JavaScript1.3">
function toRadians(degrees){
return degrees*Math.PI/180
}//end function toRadians
//============================================//
function toDegrees(radians){
return radians*180/Math.PI
}//end function toDegrees
//============================================//
var adj = 3
var hyp = 5
var ratio = adj/hyp
var angRad = Math.acos(ratio)
var angDeg = toDegrees(angRad)
document.write("radians = " + angRad + "</br>")
document.write("degrees = " + angDeg)
</script>
</body></html>
Similar to a previous script
If you examine the code in Listing 4 carefully, you will see that it is very similar to the code in Listing 2 with a couple of exceptions:
The output
When you load your html file into your browser, it should produce the output shown earlier in Figure 5 . In other words, we know that the angle at the origin didn't change. What changed was the manner in which we computed the value of that angle.
Different approaches to the same solution
In Listing 2 , we used the length of the hypotenuse and the length of the opposite side, along with the arcsine method to compute the angle.
In Listing 4 , we used the length of the hypotenuse and the length of the adjacent side, along with the arccosine method to compute the angle.
Which approach should you use?
As would be expected, since the angle didn't change, both approaches produced the same result. Deciding which approach to use often depends on the values that are available to use in the computation.
Sometimes you only have the lengths of the hypotenuse and the opposite side available, in which case you could use the arcsine. Sometimes you only have the lengths of the hypotenuse and the adjacent side available, in which case you could use the arccosine. Sometimes you have the lengths of both the opposite side and the adjacent side in addition to the length of the hypotenuse, in which case you can use either approach.
Both approaches use the length of the hypotenuse
It is important to note however that both of these approaches require you to have the length of the hypotenuse. Later in this module we will discuss the tangent and arctangent for an angle, which allows us to work with the opposite side and the adjacent side devoid of the length of the hypotenuse. (Of course, if you have the lengths of the opposite side and the adjacent side, you can always find the length of the hypotenuse using the Pythagorean theorem.)
Interesting cosine equations
The equations in Figure 8 are similar to equations in Figure 7 . The difference is that the equations in Figure 7 are based on the use of the sine of the angle and the opposite side whereas the equations in Figure 8 are based on the use of the cosine of the angle and the adjacent side.
As you can see in Figure 8 , if you know any two of the values for angle , adj , and hyp , you can find the other value. This is illustrated in the script shown in Listing 5 , which produces the output shown in Figure 9 .
Interesting cosine equations.  


Finding the length of the adjacent side
The code in Listing 5 is very similar to the code in Listing 2 . The main difference is that Listing 2 is based on the use of the sine of the angle and the length of the opposite side whereas Listing 5 is based on the use of the cosine of the angle and the length of the adjacent side.
<! File JavaScript05.html >
<html><body>
<script language="JavaScript1.3">
function toRadians(degrees){
return degrees*Math.PI/180
}//end function toRadians
//============================================//
function toDegrees(radians){
return radians*180/Math.PI
}//end function toDegrees
//============================================//
var hyp = 5
var angDeg = 53.13
var angRad = toRadians(angDeg)
var cosine = Math.cos(angRad)
var adj = hyp * cosine
document.write("adjacent = " + adj + "</br>")
hyp = adj/cosine
document.write("hypotenuse = " + hyp + "</br>")
</script>
</body></html>
No further explanation needed
Because of the similarity of Listing 5 and Listing 2 , no further explanation of the code in Listing 5 should be needed. As you can see from Figure 9 , the output values match the known lengths for the hypotenuse and the adjacent side for the triangle on your plot board.
Output for script in Listing 5.  


Computing length of adjacent side with the Google calculator
We could also compute the length of the adjacent side using the Google calculator.
The length of the adjacent side  sample computation
Enter the following into the Google search box:
5*cos(53.1301024 degrees)
The following will appear immediately below the search box:
5 * cos(53.1301024 degrees) = 3
This is the length of the adjacent side for the given angle and the given length of the hypotenuse.
Two very important equations
From an introductory physics viewpoint, two of the most important and perhaps most frequently used equations from Figure 7 and Figure 8 are shown in Figure 10 .
Two very important equations.  


These two equations are so important that it might be worth your while to memorize them. Of course, you will occasionally need most of the equations in Figure 7 and Figure 8 , so you should try to remember them, or at least know where to find them when you need them.
Vectors
As you will see later in the module that deals with vectors, you are often presented with something that resembles the hypotenuse of a right triangle whose adjacent side is on the horizontal axis and whose opposite side is parallel to the vertical axis.
The thing that looks like the hypotenuse of a right triangle is called a vector . It has a length and it has a direction. Typically, the direction is stated as the angle between the vector and the horizontal axis. Thus, the direction is analogous to the angle at the origin in the triangle on your graph board.
Horizontal and vertical components
For reasons that I won't explain until we get to that module, you will often need to compute the horizontal and vertical components of the vector. The horizontal component is essentially the adjacent side of our current right triangle. Thus, the value of the horizontal component can be computed using the second equation in Figure 10 .
The vertical component is essentially the opposite side of our current right triangle, and its value can be computed using the first equation in Figure 10 .
Once again, although the tangent of an angle is based on very specific geometric considerations involving circles (see http://www.clarku.edu/~djoyce/trig/ ), for our purposes, the tangent of an angle is simply a ratio between the lengths of two different sides of a right triangle.
A ratio of two sides
For our purposes, we will say that the tangent of an angle is equal to the ratio of the opposite side and the adjacent side. Therefore, in the case of the 345 triangle that you have on your graph board, the tangent of the angle at the origin is equal to 4/3 or 1.333.
Not limited to 1.0
Note that the absolute value for the sine and the cosine of an angle is limited to a maximum value of 1.0. However, the tangent of an angle is not so limited. In fact, the tangent of 45 degrees is 1.0 and the tangent of 90 degrees is infinity. This results from the length of the adjacent side, which is the denominator in the ratio, going to zero at 90 degrees.
Dividing by zero in a script is usually not a good thing. This is a pitfall that you must watch out for when working with tangents. I will provide code later on that shows you how deal with this issue.
Computing the tangent
If we know the lengths of the opposite side and the adjacent side, we can compute the tangent and use it for other purposes later without having to know the value of the angle.
Conversely, if we know the value of the angle but don't know the lengths of the adjacent side and/or the opposite side, we can obtain the tangent value using a scientific calculator or lookup table and use it for other purposes later.
The tangent of an angle  sample computation
Enter the following into the Google search box:
tan(53.13010235415598 degrees)
The following will appear immediately below the search box:
tan(53.13010235415598 degrees) = 1.33333333
This agrees with the ratio that we computed earlier .
The arctangent (inverse tangent) of an angle
The arctangent of an angle is the value of the angle having a given tangent value. (For example, as mentioned above, the arctangent of infinity is 90 degrees and the arctangent of 1.0 is 45 degrees.) In other words, if you know the value of the tangent of an unknown angle, you can use a scientific calculator or lookup table to find the value of the angle.
For example, we know that the tangent of the angle at the origin on your graph board is 1.333. From that, we can determine the value of the angle.
The arctangent of an angle  sample computation
Enter the following into the Google search box:
arctan(4/3) in degrees
The following will appear immediately below the search box:
arctan(4/3) = 53.1301024 degrees
We can also write a JavaScript script to perform the calculation.
Getting the angle for a known tangent value using JavaScript
Please create an html file containing the code shown in Listing 6 and open it in your browser.
<! File JavaScript06.html >
<html><body>
<script language="JavaScript1.3">
function toRadians(degrees){
return degrees*Math.PI/180
}//end function toRadians
//============================================//
function toDegrees(radians){
return radians*180/Math.PI
}//end function toDegrees
//============================================//
var opp = 4
var adj = 3
var ratio = opp/adj
var angRad = Math.atan(ratio)
var angDeg = toDegrees(angRad)
document.write("radians = " + angRad + "</br>")
document.write("degrees = " + angDeg)
</script>
</body></html>
The output from the script
Once again, when you open this file in your browser, the output shown in Figure 5 should appear in your browser window.
The code in Listing 6 is very similar to the code in Listing 2 . They both describe the same right triangle, so the output should be the same in both cases.
The code in Listing 2 uses the opposite side and the hypotenuse along with the arcsine to compute the angle. The code in Listing 6 uses the opposite side and the adjacent side along with the arctangent to compute the angle. Otherwise, no further explanation should be required.
Interesting tangent equations
In the spirit of Figure 7 and Figure 8 , Figure 11 provides some interesting equations that deal with the angle, the opposite side, and the adjacent side. Given any two, you can find the third using either the tangent or arctangent.
Interesting tangent equations.  


An exercise involving the tangent
Please copy the code from Listing 7 into an html file and open it in your browser.
<! File JavaScript07.html >
<html><body>
<script language="JavaScript1.3">
function toRadians(degrees){
return degrees*Math.PI/180
}//end function toRadians
//============================================//
function toDegrees(radians){
return radians*180/Math.PI
}//end function toDegrees
//============================================//
var adj = 3
var angDeg = 53.13
var angRad = toRadians(angDeg)
var tangent = Math.tan(angRad)
var opp = adj * tangent
document.write("opposite = " + opp + "</br>")
adj = opp/tangent
document.write("adjacent = " + adj + "</br>")
</script>
</body></html>
When you open your html file in your browser, the output shown in Figure 12 should appear in your browser window. We can see that the values in Figure 12 are correct for our 345 triangle.
Output for script in Listing 7.  


Very similar code
The code in Listing 7 is very similar to the code in Listing 3 and Listing 5 . The essential differences are that
You should be able to work through those differences without further explanation from me.
The cotangent of an angle
There is also something called the cotangent of an angle, which is simply the ratio of the adjacent side to the opposite side. If you know how to work with the tangent, you don't ordinarily need to use the cotangent, so I won't discuss it further.
Computing length of opposite side with the Google calculator
We could also compute the length of the opposite side using the Google calculator.
The length of the opposite side  sample computation
Enter the following into the Google search box:
3*tan(53.1301024 degrees)
The following will appear immediately below the search box:
3 * tan(53.1301024 degrees) = 4.00000001
Up to this point, we have dealt exclusively with angles in the range of 0 to 90 degrees (the first quadrant). As long as you stay in the first quadrant, things are relatively straightforward.
As you are probably aware, however, angles can range anywhere from 0 to 360 degrees (or more). Once you begin working with angles that are greater then 90 degrees, things become a little less straightforward.
Another svg file
When you downloaded the zip file named Phy1020.zip using the link given above , you should also have found that the zip file contains a file named Phy1020a1.svg.
The purpose of this file is to make it possible for you to create tactile graphics for sine and cosine curves using the procedure explained in the earlier module named Manual Creation of Tactile Graphics at http://cnx.org/content/m38546/latest/ .
If you have the ability to create tactile graphics, you don't need to perform the work in the following graph board exercise. However, you should read about it anyway because that will probably help you to better understand the sine and the cosine of an angle.
I will get back to tactile graphics after I describe the graph board exercise.
Another graph board exercise
In this graph board exercise, we will plot a graph of the amplitude of the sine of an angle on the vertical axis versus the angle itself on the horizontal.
We will also do the same thing for the cosine. It would be very good if you could plot one curve on the top half of your graph board and the other curve on the bottom half of your graph board so that you can easily compare the two.
Interpreting gridline values
Since I don't know how many tactile grid lines there are on your graph board, I can't tell you exactly how to interpret the grid lines so as to make maximum use of the space on the board. All that I can tell you is that the vertical amplitude values for each curve will range from 1.0 to +1.0. We would like to plot values from 360 degrees to + 360 degrees on the horizontal. You should interpret the values of the gridlines on your graph board accordingly.
Sinusoidal amplitude versus angle
Please copy the code from Listing 8 into an html file and open the file in your browser.
<;! File JavaScriptZZ.html >
<;html><;body>
<;script language="JavaScript1.3">
function toRadians(degrees){
return degrees*Math.PI/180
}//end function toRadians
//============================================//
function toDegrees(radians){
return radians*180/Math.PI
}//end function toDegrees
//============================================//
var angInc = 90
var angStart = 360
var ang = angStart
var angEnd = 360
var sine
var cosine
while(ang <;= angEnd){
//Compute sine and cosine of angle
sine = Math.sin(toRadians(ang))
cosine = Math.cos(toRadians(ang))
//Reduce the number of digits in the output
sine = (Math.round(100*sine))/100
cosine = (Math.round(100*cosine))/100
//Display the results
document.write("Angle: " + ang +
" Sine: " + sine +
" Cosine: " + cosine +
"<;/br>")
//Increase the angle for next iteration
ang = ang + angInc
}//end while loop
<;/script>
<;/body><;/html>
Output from the script
When you open your html file in your browser, the output shown in Figure 13 should appear in your browser window.
Sinusoidal values at 90degree increments.  


Figure 13 contains the data for two different curves. One is a sine curve and the other is a cosine curve.
Plot the points using pushpins
You should be able to plots these data values as two separate curves on your graph board by inserting pushpins at the coordinate values shown and then connecting the pushpins with rubber bands, pipe cleaners, yarn, flexible wire, or something similar. (Rubber bands might not work if you are using a homemade plot board constructed from Styrofoam, because the pins pull too easily.)
Remember, the angle values from 360 degrees (2*PI radians) to +360 degrees (+2*PI radians) are horizontal coordinates while the corresponding values for the sine and cosine are vertical coordinates.
Saw tooth curves
Once you have plotted the points, you should be able to discern two curves, each of which is a saw tooth.
The two curves have exactly the same shape, but one is shifted horizontally relative to the other. For example, the sine curve has a value of zero at an angle of zero (the origin) and it is asymmetric about the vertical axis.
The cosine curve, on the other hand has a value of 1 at an angle of zero and it is symmetric about the vertical axis.
Periodic curves
These are periodic curves. For example, the shape of the sine curve between 360 and 0 is the same as the shape of the sine curve between 0 and +360. Each of those ranges represents one cycle of the periodic curve.
We only computed the values from 360 to +360. However, if we had computed the values from 3600 to + 3600, the overall shape of the curve would not differ from what we have here. The shape of each cycle of the curve would be identical to the shape of the cycle to the left and the cycle to the right.
Not really a saw tooth
The sine and cosine curves don't really have a saw tooth shape. That is an artifact of the fact that we didn't compute enough points to reliably describe the shape of the curves. Let's improve on that.
Modify the script
Modify the code in your script to initialize the value of the variable named angInc to 45 degrees instead of 90 degrees and then load the revised version into your browser. This will cause the script to fill in data points between the points that we already have producing the output shown in Figure 16 .
Sinusoidal values at 45degree increments.  


Plot the new points
Every other line of text in Figure 14 should contain sine and cosine values for angles that are half way between the points that you already have plotted. Use pushpins to plot the new points and connect all of the points in each curve using rubber bands, pipe cleaners, or whatever you find most useful for this purpose.
Same shape but shifted horizontally
The two curves still have the same shape, although shifted horizontally relative to one another and they are still periodic. However, they no longer have a saw tooth shape. They tend to be a little more rounded near the peaks and they are beginning to provide a better representation of the actual shapes of the sine and cosine curves.
Let's do it again
Change the value of the variable named angInc from 45 degrees to 22.5 degrees and load the new version of the html file into your browser. Now the output should look like Figure 15 .
Sinusoidal values at 22.5degree increments.  


A lot of data points
Once again, every other line of text in Figure 15 contains new sine and cosine values for angles that you don't have plotted yet.
Plotting all of these point is going to require a lot of pushpins and a lot of effort. Before you do that, let's think about it.
Two full cycles
You should have been able to discern by now that your plots for the sine and cosine graphs each contain two full cycles. An important thing about periodic functions is that once you know the shape of the curve for any one cycle, you know the shape of the curve for every cycle from minus infinity to infinity. The shape of every cycle is exactly the same as the shape of every other cycle.
Saving pushpins and effort
If you are running out of pushpins or running out of patience, you might consider updating your plots for only one cycle.
You should be able to discern that your curves no longer have a saw tooth shape. Each time we have run the script, we have sampled the amplitude values of each curve at twice as many points as before. Therefore, the curves should be taking on a smoother rounded shape that is better representation of the actual shape of the curves.
Continue the process
You can continue this process of improving the curves for as long as you have the graph board space, pushpins, and patience to do so. Just divide the value of the variable named angInc by a factor of two and rerun the script. That will produce twice as many data points that are only half as far apart on the horizontal axis.
If you choose to do so, you can plot only the new points in onehalf of a cycle to get an idea of the shape. By now you should have discerned that each half of a cycle has the same shape, only one half is above the horizontal axis and the other half is a mirror image below the axis.
Plot of cosine and sine curves
Getting back to tactile graphics, for the benefit of any sighted person that may be assisting you, Figure 16 shows a cosine curve plotted above a sine curve very similar to the curves that you have plotted on your graph board. ( Figure 16 shows a mirror image of the actual curves because the file named Phy1020a1.svg is intended to be used for manual embossing from the back of the paper.) A nonmirrorimage version is shown in Figure 20 .
Mirror image plot of cosine and sine curves from the file named Phy1020a1svg. 

This image contains lines that aren't straight, but with a little care, your assistant should be able to do a reasonably good job of embossing them when the image is printed as the full 8.5x11 inch version.
Page setup
If you use the IVEO Viewer to print the file named Phy1020a1.svg, you should use the default Page Setup selection for Letter (Landscape).
Grid lines
The image in Figure 16 contains 7 vertical grid lines. The vertical grid line in the center represents an angle of zero degrees. The space between each grid line on either side of the center represents an angle of 90 degrees or PI/2 radians.
There are two horizontal grid lines. One is onefourth of the way down from the top. The other is onefourth of the way up from the bottom.
The curves
A cosine curve is plotted with vertical values relative to the top grid line. It extends from 360 degrees on the left to +360 degrees on the right.
A sine curve is plotted with vertical values relative to the bottom grid line. It also extends from 360 degrees on the left to +360 degrees on the right. (Note once again that Figure 16 was flipped horizontally to crate a mirror image.)
Return values for the Math.asin, Math.acos, and Math.atan methods
I told you earlier that the Math.asin method returns a value between PI/2 and PI/2. However, I didn't tell you that the Math.acos method returns a value between 0 and PI, or that the Math.atan method returns a value between PI/2 and PI/2. You now have enough information to understand why this is true.
Smooth curves
If you examine the two curves that you have just plotted, you can surmise that the sine and cosine functions are smooth curves whose values range between 1 and +1 inclusive. For every possible value between 1 and +1, there is an angle in the range PI/2 and PI/2 whose sine value matches that value. There is also an angle in the range 0 and PI whose cosine value matches that value.
(Although you haven't plotted the curve for the tangent, a similar situation holds there also.)
An infinite number of angles
Therefore, given a specific numeric value between 1 and +1, there are an infinite number of angles whose sine and cosine values match that numeric value and the method has no way of distinguishing between them. Therefore, the Math.asin method returns the matching angle that is closest to zero and the Math.acos method returns the matching positive angle that is closest to zero.
What can we learn from this?
One important thing that we can learn is there is no difference between the sine or cosine of an angle and the sine or cosine of a different angle that differs from the original angle by 360 degrees. Thus, the Math.asin and Math.acos methods cannot be used to distinguish between angles that differ by 360 degrees. (As you learned above, the situation involving the Math.asin and Math.acos methods is even more stringent than that.)
Onequarter cycle contains all of the information
Another thing that we can learn is that once you know the shape of the cosine curve from 0 degrees to 90 degrees, you have enough information to construct the entire cosine curve and the entire sine curve across any range of angles. Every possible value or the negative of every possible value that can occur in a sine or cosine curve occurs in the cosine curve between 0 degrees and 90 degrees. Furthermore, the order of those values is also well defined.
Think about these relationships
You should think about these kinds of relationships. As I mentioned earlier, as long as we are working with angles between 0 and 90 degrees, everything is relatively straightforward. However, once we start working with angles between 90 degrees and 360 degrees (or greater), things become a little less straightforward.
If you have a good picture in your mind of the shape of the two curves between 360 degrees and +360 degrees, you may be able to avoid errors once you start working on physics problems that involve angles outside the range of 0 to 90 degrees.
Quadrants
We often think of a twodimensional space with horizontal and vertical axes and the origin at the center in quadrants. Each quadrant is bounded by half the horizontal axis and half the vertical axis.
It is common practice to number the quadrants in counterclockwise order with the upperright quadrant being quadrant 1, the upperleft quadrant being quadrant 2, the bottomleft quadrant being quadrant 3, and the bottomright quadrant being quadrant 4.
Angles fall in quadrants
If you measure the angle between the positive horizontal axis and a line segment that emanates from the origin, quadrant 1 contains angles between 0 and PI/2, quadrant 2 contains the angles between PI/2 and PI, quadrant 3 contains the angles between PI and 3*PI/2, and quadrant 4 contains the angles between 3*PI/2 and 2*PI (or zero). (Note that I didn't attempt to reconcile the inclusion of each axis in the two quadrants on either side of the axis.)
Algebraic signs versus quadrant number
It is sometimes useful to consider how the algebraic sign of the sine, cosine, and tangent values varies among the four quadrants. Figure 17 contains a table that shows the sign of the sine, cosine, and tangent values for each of the four quadrants
Algebraic signs versus quadrants.  


Working with arctangents is more difficult than arcsine or arccosine
Working with arctangent is somewhat more difficult than working with arcsine or arccosine, if for no other reason than the possibility of dividing by zero when working with the arctangent.
Listing 9 shows a JavaScript function named getAngle that deals with this issue.
<! File JavaScript09.html >
<html><body>
<script language="JavaScript1.3">
document.write("Start Script </br>");
//The purpose of this function is to receive the adjacent
// and opposite side values for a right triangle and to
// return the angle in degrees in the correct quadrant.
function getAngle(adjacent,opposite){
if((adjacent == 0) && (opposite == 0)){
//Angle is indeterminate. Just return zero.
return 0;
}else if((adjacent == 0) && (opposite > 0)){
//Avoid divide by zero denominator.
return 90;
}else if((adjacent == 0) && (opposite < 0)){
//Avoid divide by zero denominator.
return 90;
}else if((adjacent < 0) && (opposite >= 0)){
//Correct to second quadrant
return Math.atan(opposite/adjacent)*180/Math.PI + 180;
}else if((adjacent < 0) && (opposite <= 0)){
//Correct to third quadrant
return Math.atan(opposite/adjacent)*180/Math.PI + 180;
}else{
//First and fourth quadrants. No correction required.
return Math.atan(opposite/adjacent)*180/Math.PI;
}//end else
}//end function getAngle
//Modify these values and run for different cases.
var adj = 3;
var opp = 4;
document.write("adj = " + adj.toFixed(2) +
" opp = " + opp.toFixed(2) + " units</br>");
document.write("angle = " + getAngle(adj,opp).toFixed(2)
+ " units</br>");
var adj = 3;
var opp = 4;
document.write("adj = " + adj.toFixed(2) +
" opp = " + opp.toFixed(2) + " units</br>");
document.write("angle = " + getAngle(adj,opp).toFixed(2)
+ " units</br>");
var adj = 3;
var opp = 4;
document.write("adj = " + adj.toFixed(2) +
" opp = " + opp.toFixed(2) + " units</br>");
document.write("angle = " + getAngle(adj,opp).toFixed(2)
+ " units</br>");
var adj = 3;
var opp = 4;
document.write("adj = " + adj.toFixed(2) +
" opp = " + opp.toFixed(2) + " units</br>");
document.write("angle = " + getAngle(adj,opp).toFixed(2)
+ " units</br>");
</script>
</body></html>
The code in Listing 9 begins by defining a function named getAngle that accepts the signed values of the adjacent side and the opposite side of the right triangle and returns the angle that the hypotenuse makes with the positive horizontal axis.
Then the code in Listing 9 tests the result for four different triangles situated in each of the four quadrants.
Figure 18 shows the output produced by this script.
Output from the code in Listing 9.  


This is an issue that will become important when we reach the module that deals with vectors in all four quadrants.
Structure of the script
The script shown in Listing 9 begins by defining a function named getAngle . The purpose of this function is to return an angle in degrees in the correct quadrant based on the lengths of the adjacent and opposite sides of an enclosing right triangle. As mentioned above, the returned angle is the angle that the hypotenuse makes with the positive horizontal axis.
An indeterminate result
The getAngle function calls the Math.atan method to compute the angle whose tangent is the ratio of the opposite side to the adjacent side of a right triangle.
If the lengths of both the opposite and adjacent sides are zero, the ratio opposite/adjacent is indeterminate and the value of the angle cannot be computed. In fact there is no angle corresponding to the ratio 0/0. However, the function must either return the value of an angle, or must return some sort of flag indicating that computation of the angle is not possible.
In this case, the function simply returns the value zero for the angle.
Avoiding division by zero
If the length of adjacent side is zero and the length of opposite side is not zero, the ratio opposite/adjacent is infinite. Therefore, the value of the angle cannot be computed. However, in this case, the angle is known to be 90 degrees (for opposite greater than zero) or 270 degrees (90 degrees, for opposite less than zero). The getAngle function traps both of those cases and returns the correct angle in each case.
Correcting for the quadrant
The Math.atan method receives one parameter and it is either a positive or negative value. If the value is positive, the method returns an angle between 0 and 90 degrees. If the value is negative, the method returns an angle between 0 and 90 degrees. Thus, the angles returned by the Math.atan method always lie in the first or fourth quadrants.
(Actually, as I mentioned earlier, +90 degrees and 90 degrees are not possible because the tangent of +90 degrees or 90 degrees is an infinitely large positive or negative value. However, the method can handle angles that are very close to +90 or 90 degrees.)
A negative opposite/adjacent ratio
If the opposite/adjacent ratio is negative, this doesn't necessarily mean that the angle lies in the fourth quadrant. That negative ratio could result from a positive value for opposite and a negative value for adjacent. In that case, the angle would lie in the second quadrant between 90 degrees and 180 degrees.
The getAngle function tests the signs of the values for opposite and adjacent. If the signs indicate that the angle lies in the second quadrant, the value returned from the Math.atan method is corrected to place the angle in the second quadrant. The corrected angle is returned by the getAngle function.
A positive opposite/adjacent ratio
Similarly, if the opposite/adjacent ratio is positive, this doesn't necessarily mean that the angle lies in the first quadrant. That positive ratio could result from a negative opposite value and a negative adjacent value. In that case, the angle would lie in the third quadrant between 180 degrees and 270 degrees.
Again, the getAngle function tests the signs of the values for opposite and adjacent. If both values are negative, the value returned from the Math.atan method is corrected to place the angle in the third quadrant.
No corrections required...
Finally, if no corrections are required for the quadrant, the getAngle function returns the value returned by the Math.atan method. Note however, that in all cases, the Math.atan method returns the angle in radians. That value is converted to degrees by the getAngle function and the returned value is in degrees.
Positive and negative angles
As you can see from the results of the test shown in Figure 18 , angles in the first, second, and third quadrants are returned as positive angles in degrees. However, angles in the fourth quadrant are returned as negative angles in degrees.
Figure 2 and Figure 16 show the actual mirror image versions of the images contained in the svg files. Figure 19 and Figure 20 show the same images in their nonmirrorimage orientation.
Normal image from file Phy1020b1.svg. 

Normal image plot of cosine and sine curves from the file named Phy1020a1svg. 

I encourage you to run the scripts that I have presented in this lesson to confirm that you get the same results. Copy the code for each script into a text file with an extension of html. Then open that file in your browser. Experiment with the code, making changes, and observing the results of your changes. Make certain that you can explain why your changes behave as they do.
I will publish a module containing consolidated links to resources on my Connexions web page and will update and add to the list as additional modules in this collection are published.
This section contains a variety of miscellaneous information.
Financial : Although the Connexions site makes it possible for you to download a PDF file for this module at no charge, and also makes it possible for you to purchase a preprinted version of the PDF file, you should be aware that some of the HTML elements in this module may not translate well into PDF.
I also want you to know that I receive no financial compensation from the Connexions website even if you purchase the PDF version of the module.
Affiliation : I am a professor of Computer Information Technology at Austin Community College in Austin, TX.
end
"Blind students should not be excluded from physics courses because of inaccessible textbooks. The modules in this collection present physics concepts in a format that blind students can read […]"