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Quadrants
We often think of a two-dimensional space with horizontal and vertical axes and the origin at the center in quadrants. Each quadrant is bounded byhalf the horizontal axis and half the vertical axis.
It is common practice to number the quadrants in counter-clockwise order with the upper-right quadrant beingquadrant 1, the upper-left quadrant being quadrant 2, the bottom-left quadrant being quadrant 3, and the bottom-right quadrant being quadrant 4.
Angles fall in quadrants
If you measure the angle between the positive horizontal axis and a line segment thatemanates 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 theangles 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 ofeach 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 ofthe four quadrants
Figure 17 . Algebraic signs versus quadrants. |
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1 2 3 4
sine + + - -cosine + - - +
tangent + - + - |
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 zerowhen working with the arctangent.
Listing 9 shows a JavaScript function named getAngle that deals with this issue.
Listing 9 . A function to deal with quadrants. |
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<!---------------- 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 quadrantreturn Math.atan(opposite/adjacent)*180/Math.PI + 180;
}else if((adjacent<0)&&(opposite<= 0)){
//Correct to third quadrantreturn 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> |
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