A triangle has two corners with angles of $\frac{π}{6}$ $pi / 6$ and $\frac{5 π}{8}$ $(5 pi )/ 8$ . If one side of the triangle has a length of $8$ $8$ , what is the largest possible area of the triangle?

Question

A triangle has two corners with angles of $\frac{π}{6}$ $pi / 6$ and $\frac{5 π}{8}$ $(5 pi )/ 8$ . If one side of the triangle has a length of $8$ $8$ , what is the largest possible area of the triangle?

1 Answer

Impact of this question

10436 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-19T05:01:41

Area of triangle $A_{t} = (\frac{1}{2}) a b sin C \approx 36$ $A_t = (1/2) a b sin C ~~ color (red)(36$

Explanation:

enter image source here

Given $ˆ A = \frac{π}{6}, ˆ B \frac{5 π}{8}$ $hatA = pi/6, hat B (5pi)/8$ , one side = 8#

To find the largest possible area of the triangle.

Third angle $ˆ C = π - \frac{π}{6} - \frac{5 π}{8} = \frac{5 π}{24}$ $hatC = pi - pi/6 - (5pi)/8 = (5pi)/24$

To get the largest possible area, side8 should correspond to to the least angle.

$\frac{a}{sin (\frac{5 π}{24})} = \frac{b}{sin (\frac{5 π}{8})} = \frac{c}{sin (π)} / 6$ $a/ sin ((5pi)/24) = b / sin ((5pi)/8) = c / sin (pi)/6$ using, law of sines.

$a = \frac{8 \cdot sin (\frac{5 π}{24})}{sin (\frac{π}{6})} = 9.7402$ $a = (8 * sin ((5pi)/24)) / sin (pi/6) = 9.7402$

$b = \frac{8 \cdot sin (\frac{5 π}{8})}{sin (\frac{π}{6})} = 14.7821$ $b = (8 * sin ((5pi)/8)) / sin (pi/6) = 14.7821$

Area of triangle $A_{t} = (\frac{1}{2}) a b sin C = (\frac{1}{2}) \cdot 9.7402 \cdot 14.7821 \cdot sin (\frac{π}{6}) \approx 36$ $A_t = (1/2) a b sin C = (1/2) * 9.7402 * 14.7821 * sin (pi/6) ~~ color (red)(36$

Science

Math

Humanities

... and beyond

A triangle has two corners with angles of $\frac{π}{6}$ $pi / 6$ and $\frac{5 π}{8}$ $(5 pi )/ 8$ . If one side of the triangle has a length of $8$ $8$ , what is the largest possible area of the triangle?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

A triangle has two corners with angles of π6 pi / 6 and 5π8 (5 pi )/ 8 . If one side of the triangle has a length of 88 , what is the largest possible area of the triangle?

1 Answer

Explanation:

Related questions

Impact of this question

A triangle has two corners with angles of $\frac{π}{6}$ $pi / 6$ and $\frac{5 π}{8}$ $(5 pi )/ 8$ . If one side of the triangle has a length of $8$ $8$ , what is the largest possible area of the triangle?