A triangle has two corners with angles of $pi / 12$ and $(5 pi )/ 8$ . If one side of the triangle has a length of $1$ , what is the largest possible area of the triangle?

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Answer 1 · 2017-12-05T09:22:42

Area of largest possible triangle $= color (red)(2.0056)$

Three angles are $pi/12, (5pi)/8, (pi - (pi/12) + ((5pi)/8) =( 7pi)/24$

$a/ sin A = b / sin B = c / sin C$

To get the largest possible are, smallest angle should correspond to the side of length 1.

$1 / sin (pi/12) = b / sin ((7pi)/24) = c / sin ((5pi)/8)$

$b = (sin ((7pi)/24)) / (sin (pi/12)$
$b = 3.0653$

$c = (sin ((5pi)/8)) / (sin (pi/12))$
$c = 3.5696$

Semi perimeter $s = (a + b + c) / 2 = (1+3.0653+3.5696)/2 = 3.8175$

$s-a = 3.8175-1 = 2.8175$
$s-b = 3.8175-3.0653 = 0.7522$
$s-c = 3.8175-3.5696 = 0.2479$

Area of $Delta = sqrt(s (s-a) (s-b) (s-c))$

$Area of Delta = sqrt(3.8175 * 2.8175 * 0.7522 * 0.2479) = color (red)(2.0056)$

A triangle has two corners with angles of pi / 12 pi / 12 and (5 pi )/ 8 (5 pi )/ 8 . If one side of the triangle has a length of 1 1 , what is the largest possible area of the triangle?