Question 935359
Two sides of a parallelogram are 570 feet and 410 feet. The measure of the angle between these sides is 166 degrees. Find the area of the parallelogram to the nearest square foot.
<pre>
{{{drawing(400,4000/53,-50,1010,-50,150,arc(0,0,300,-300,0,14)  ,
locate(155,95,410),
line(0,0,397.8212478,99.1879772),

line(397.8212478,99.1879772,967.8212478,99.1879772),

line(967.8212478,99.1879772,570,0),
line(570,0,0,0),
locate(400,90,"166°"),locate(150,45,"14°"),
locate(600,140,570),
red(arc(397.8212478,99.1879772,50,-50,194,360)),
locate(285,0,570)



  )}}}

The given angle (upper left) is 166° and since two 
adjacent angles of any parallelogram are supplementary
the angle at the lower left is 180°-166° = 14°

To find the area of the parallelogram, we need to find
it's height.  So we draw in the green line for the height
and label it h:

{{{drawing(400,4000/53,-50,1010,-50,150,arc(0,0,300,-300,0,14)  ,
locate(155,95,410),
line(0,0,397.8212478,99.1879772),

line(397.8212478,99.1879772,967.8212478,99.1879772),

line(967.8212478,99.1879772,570,0),locate(370,70,h),
line(570,0,0,0), green(line(397.8212478,0,397.8212478,99.1879772)), 
locate(400,90,"166°"),locate(150,45,"14°"),
locate(600,140,570), locate(285,0,570),
red(arc(397.8212478,99.1879772,50,-50,194,360)) )}}}

Then 

{{{sin("14°")=OPPOSITE/HYPOTENUSE}}}

{{{sin("14°")=h/410}}}

Put a 1 under the sine

{{{sin("14°")/1=h/410}}}

Cross-multiply:

{{{h=410sin("14°")}}}

{{{h=99.1879772}}} feet.

Area of the parallelogram = base × height

                          = 570 ft × 99.1879772 ft.

                          = 56537.147 square feet

                          = 56537 square feet to the nearest square foot.

Edwin</pre>