A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $pi/6$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 5, what is the area of the triangle?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $pi/6$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 5, what is the area of the triangle?

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Answer 1 · 2016-09-07T23:45:52

Area $\approx 2.2877$ $~~ 2.2877$

Explanation:

If $∠ B C = \frac{π}{12}$ $/_BC=pi/12$ and $∠ A B = \frac{π}{6}$ $/_AB=pi/6$
then
$XXX ∠ A C = π - (\frac{π}{12} + \frac{π}{6}) = \frac{3 π}{4}$ $color(white)("XXX")/_AC = pi- (pi/12+pi/6) =(3pi)/4$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

By the Law of Sines:
$XXX \frac{A}{sin (∠ B C)} = \frac{B}{sin (∠ A C)} = \frac{C}{sin (∠ A B)}$ $color(white)("XXX")A/sin(/_BC)=B/sin(/_AC)=C/sin(/_AB)$

With the given values:
$XXX \frac{A}{sin (\frac{π}{12})} = \frac{5}{sin (\frac{3 π}{4})} = \frac{C}{sin (\frac{π}{6})}$ $color(white)("XXX")A/sin(pi/12)=5/sin((3pi)/4)=C/sin(pi/6)$

So
$XXX A = \frac{5}{sin (\frac{3 π}{4})} \cdot sin (\frac{π}{12}) \approx 1.830127019$ $color(white)("XXX")A=5/sin((3pi)/4)*sin(pi/12)~~1.830127019$
and
$XXX C = \frac{5}{sin (\frac{3 π}{4})} \cdot sin (\frac{π}{6}) \approx 3.535533906$ $color(white)("XXX")C=5/sin((3pi)/4)*sin(pi/6)~~3.535533906$

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The semi-perimeter of the triangle, $s$ $s$ , is
$XXX s = \frac{A + B + C}{2} \approx 5.182830462$ $color(white)("XXX")s=(A+B+C)/2~~5.182830462$

By Heron's Formula, the Area of the Triangle is
$XXX A = \sqrt{s (s - A) (s - B) (s - C)}$ $color(white)("XXX")A=sqrt(s(s-A)(s-B)(s-C))$

$XXXX \approx 2.287658774$ $color(white)("XXXX")~~2.287658774$

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $pi/6$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 5, what is the area of the triangle?

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Explanation:

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A triangle has sides A, B, and C. The angle between sides A and B is π6pi/6 and the angle between sides B and C is π12pi/12. If side B has a length of 5, what is the area of the triangle?

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{6}$ $pi/6$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 5, what is the area of the triangle?