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

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

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Answer 1 · 2017-12-05T09:22:42

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

Explanation:

Three angles are $\frac{π}{12}, \frac{5 π}{8}, (π - (\frac{π}{12}) + (\frac{5 π}{8}) = \frac{7 π}{24}$ $pi/12, (5pi)/8, (pi - (pi/12) + ((5pi)/8) =( 7pi)/24$

$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ $a/ sin A = b / sin B = c / sin C$

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

$\frac{1}{sin (\frac{π}{12})} = \frac{b}{sin (\frac{7 π}{24})} = \frac{c}{sin (\frac{5 π}{8})}$ $1 / sin (pi/12) = b / sin ((7pi)/24) = c / sin ((5pi)/8)$

$b = \frac{sin (\frac{7 π}{24})}{sin (\frac{π}{12})}$ $b = (sin ((7pi)/24)) / (sin (pi/12)$
$b = 3.0653$ $b = 3.0653$

$c = \frac{sin (\frac{5 π}{8})}{sin (\frac{π}{12})}$ $c = (sin ((5pi)/8)) / (sin (pi/12))$
$c = 3.5696$ $c = 3.5696$

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

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

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

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

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

1 Answer

Explanation:

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A triangle has two corners with angles of pi / 12 π12 pi / 12 and (5 pi )/ 8 5π8 (5 pi )/ 8 . If one side of the triangle has a length of 1 11 , what is the largest possible area of the triangle?

1 Answer

Explanation:

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