A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $(pi)/6$ . If side C has a length of $8$ $8$ and the angle between sides B and C is $\frac{7 π}{12}$ $(7pi)/12$ , what are the lengths of sides A and B?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $(pi)/6$ . If side C has a length of $8$ $8$ and the angle between sides B and C is $\frac{7 π}{12}$ $(7pi)/12$ , what are the lengths of sides A and B?

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Answer 1 · 2018-06-01T11:54:41

$a = 4 \cdot \sqrt{2} \cdot (1 + \sqrt{3})$ $a=4*sqrt(2)*(1+sqrt(3))$ , $b = 8 \sqrt{2}$ $b=8sqrt(2)$

Explanation:

The third angle is given by $π - 9 \frac{π}{12} = \frac{π}{4}$ $pi-9pi/12=pi/4$

By the Theorem of sines we get

$\frac{sin (\frac{π}{6})}{sin (7 \cdot \frac{π}{12})} = \frac{8}{a}$ $sin(pi/6)/sin(7*pi/12)=8/a$
so

$a = 8 \cdot \frac{sin (7 \cdot \frac{π}{12})}{sin (\frac{π}{6})}$ $a=8*sin(7*pi/12)/sin(pi/6)$

and
$\frac{sin (\frac{π}{4})}{sin (\frac{π}{6})} = \frac{b}{a}$ $sin(pi/4)/sin(pi/6)=b/a$

so
b=8*sin(pi/4)/sin(pi/6)#

note that
$sin (7 \cdot \frac{π}{12}) = \frac{1}{4} \cdot \sqrt{2} \cdot (1 + \sqrt{3})$ $sin(7*pi/12)=1/4*sqrt(2)*(1+sqrt(3))$
$sin (\frac{π}{6}) = \frac{1}{2}$ $sin(pi/6)=1/2$
$sin (\frac{π}{4}) = \frac{\sqrt{2}}{2}$ $sin(pi/4)=sqrt(2)/2$

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $(pi)/6$ . If side C has a length of $8$ $8$ and the angle between sides B and C is $\frac{7 π}{12}$ $(7pi)/12$ , what are the lengths of sides A and B?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $(pi)/6$ . If side C has a length of $8$ $8$ and the angle between sides B and C is $\frac{7 π}{12}$ $(7pi)/12$ , what are the lengths of sides A and B?