Question 282780
Two tow trucks try to pull a car out of a ditch. One applies a force of 1500 lbs, the other 2000lbs. The resultant force is 3000lbs. What is the angle between them?
<pre><font size = 4 color = "indigo"><b>

The other tutor's solution is incorrect!

Here are the two forces.  We want to find {{{theta}}}

{{{drawing(400,400,-1,5,-.2,6.2,

line(3cos(30*pi/180),3sin(30*pi/180),0,0),
locate(-.5,4.3,"2000#"),
line(4cos(92.72038726*pi/180),4sin(92.72038726*pi/180),0,0)  ,
locate(.1,.5,theta),
line(4cos(92.72038726*pi/180)-.2,
     4sin(92.72038726*pi/180)-.2,
     4cos(92.72038726*pi/180),
     4sin(92.72038726*pi/180) ),

line(4cos(92.72038726*pi/180)+.2,
     4sin(92.72038726*pi/180)-.2,
     4cos(92.72038726*pi/180),
     4sin(92.72038726*pi/180) ),

locate(2.7,1.8,"1500#"),

line(3cos(30*pi/180)-.1,
     3sin(30*pi/180)-.4,
     3cos(30*pi/180),
     3sin(30*pi/180) ),

line(3cos(30*pi/180)-.3,
     3sin(30*pi/180)+.2,
     3cos(30*pi/180),
     3sin(30*pi/180) )

   )}}}

To draw the resultant, we first draw 2 lines, one 
through each arrowhead parallel to the other vector.
This completes a parallelogram.
I'll draw them in green below.  Notice that the angle 
labeled {{{alpha}}} is the supplement of angle {{{theta}}}

{{{drawing(400,400,-1,5,-.2,6.2,

line(3cos(30*pi/180),3sin(30*pi/180),0,0),
locate(-.5,4.3,"2000#"),
line(4cos(92.72038726*pi/180),4sin(92.72038726*pi/180),0,0)  ,
locate(.1,.5,theta),
line(4cos(92.72038726*pi/180)-.2,
     4sin(92.72038726*pi/180)-.2,
     4cos(92.72038726*pi/180),
     4sin(92.72038726*pi/180) ),
locate(0.1,4,alpha),
line(4cos(92.72038726*pi/180)+.2,
     4sin(92.72038726*pi/180)-.2,
     4cos(92.72038726*pi/180),
     4sin(92.72038726*pi/180) ),

locate(2.7,1.8,"1500#"),

line(3cos(30*pi/180)-.1,
     3sin(30*pi/180)-.4,
     3cos(30*pi/180),
     3sin(30*pi/180) ),

line(3cos(30*pi/180)-.3,
     3sin(30*pi/180)+.2,
     3cos(30*pi/180),
     3sin(30*pi/180) ),

green(line(4cos(92.72038726*pi/180)+3cos(30*pi/180),
     4sin(92.72038726*pi/180)+3sin(30*pi/180),

   4cos(92.72038726*pi/180),
     4sin(92.72038726*pi/180) )),

green(line(4cos(92.72038726*pi/180)+3cos(30*pi/180),
     4sin(92.72038726*pi/180)+3sin(30*pi/180),

   3cos(30*pi/180),
     3sin(30*pi/180) ))

   )}}} 

Then the resultant is the vector whose tail coincides
with the two vectors, and whose arrow tip is at the
opposite vertex of the parallelogram.  I'll draw the
resultant in red.  We are told that this resultant force
is 3000#:
   
{{{drawing(400,400,-1,5,-.2,6.2,

line(3cos(30*pi/180),3sin(30*pi/180),0,0),
locate(-.5,4.3,"2000#"),
line(4cos(92.72038726*pi/180),4sin(92.72038726*pi/180),0,0)  ,
locate(.1,.5,theta), red(locate(1.4,3,"3000#")),
line(4cos(92.72038726*pi/180)-.2,
     4sin(92.72038726*pi/180)-.2,
     4cos(92.72038726*pi/180),
     4sin(92.72038726*pi/180) ),
locate(0.1,4,alpha),
line(4cos(92.72038726*pi/180)+.2,
     4sin(92.72038726*pi/180)-.2,
     4cos(92.72038726*pi/180),
     4sin(92.72038726*pi/180) ),

locate(2.7,1.8,"1500#"),

line(3cos(30*pi/180)-.1,
     3sin(30*pi/180)-.4,
     3cos(30*pi/180),
     3sin(30*pi/180) ),

line(3cos(30*pi/180)-.3,
     3sin(30*pi/180)+.2,
     3cos(30*pi/180),
     3sin(30*pi/180) ),

green(line(4cos(92.72038726*pi/180)+3cos(30*pi/180),
     4sin(92.72038726*pi/180)+3sin(30*pi/180),

   4cos(92.72038726*pi/180),
     4sin(92.72038726*pi/180) )),

green(line(4cos(92.72038726*pi/180)+3cos(30*pi/180),
     4sin(92.72038726*pi/180)+3sin(30*pi/180),

   3cos(30*pi/180),
     3sin(30*pi/180) )),

red(line(4cos(92.72038726*pi/180)+3cos(30*pi/180),
     4sin(92.72038726*pi/180)+3sin(30*pi/180),0,0))

   )}}}




We have all three sides of the triangle with the angle {{{alpha}}}.
So we can use the law of cosines to find angle {{{alpha}}}

{{{cos(alpha)=(1500^2+2000^2-3000^2)/(2*1500*2000)}}}

{{{cos(alpha)=-4583333333}}}

{{{alpha="117.2796127°"}}}

Since {{{theta}}} and {{{alpha}}} are supplementary,

{{{theta="180°"-alpha="180°"-"117.2796127°"="62.72038726°"}}}

Edwin</pre>