A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{4}$ $pi/4$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 4, 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 $\frac{π}{4}$ $pi/4$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 4, what is the area of the triangle?

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Answer 1 · 2017-03-16T01:21:31

Area=1.73 $u n i t s^{2}$ $units^2$

Explanation:

First you can figure out the last angle of the triangle which is $\frac{2 π}{3}$ $(2pi)/3$ (if it helps better the angles are $15, 45, and 120$ $15, 45, and 120$ degrees.)
then you can use the law of sines to figure out another side.
$\frac{sin (\frac{2 π}{3})}{4} = \frac{sin (\frac{π}{4})}{x}$ $sin((2pi)/3)/4=sin(pi/4)/x$
$x = \frac{4 sin (\frac{π}{4})}{sin (\frac{2 π}{3})} = 3.27$ $x=(4sin(pi/4))/sin((2pi)/3)=3.27$
then do that again to find the last side
$\frac{sin (\frac{2 π}{3})}{4} = \frac{sin (\frac{π}{12})}{x}$ $sin((2pi)/3)/4=sin(pi/12)/x$
$x = \frac{4 sin (\frac{π}{12})}{sin (\frac{2 π}{3})} = 1.20$ $x=(4sin(pi/12))/sin((2pi)/3)=1.20$
then you can use Heron's formula.
First find S
$S = \frac{a + b + c}{2} = 4.24$ $S=(a+b+c)/2=4.24$
then substitute
$A = \sqrt{s (s - a) (s - b) (s - c)}$ $A=sqrt(s(s-a)(s-b)(s-c))$
$A = \sqrt{4.24 (4.24 - 1.20) (4.24 - 4) (4.24 - 3.27)} = 1.73$ $A=sqrt(4.24(4.24-1.20)(4.24-4)(4.24-3.27))=1.73$

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{4}$ $pi/4$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 4, 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 π4pi/4 and the angle between sides B and C is π12pi/12. If side B has a length of 4, 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 $\frac{π}{4}$ $pi/4$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 4, what is the area of the triangle?