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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 .

Figure 8 . Interesting cosine equations.
cosine(angle) = adj/hyp angle = arccosine(adj/hyp)adj = hyp * cosine(angle) hyp = adj/cosine(angle)

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.

Listing 5 . Finding 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 = 53.13var angRad = toRadians(angDeg) var cosine = Math.cos(angRad)var adj = hyp * cosine document.write("adjacent = " + adj + "</br>") hyp = adj/cosinedocument.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 sidefor the triangle on your plot board.

Figure 9 . Output for script in Listing 5.
adjacent = 3.0000071456633126 hypotenuse = 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 .

Figure 10 . Two very important equations.
opp = hyp * sine(angle) adj = hyp * cosine(angle)

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 equationsin 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 righttriangle whose adjacent side is on the horizontal axis and whose opposite side is parallel to the vertical axis.

Questions & Answers

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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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Nasa has use it in the 60's, copper as water purification in the moon travel.
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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