A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{3 π}{8}$ $(3pi)/8$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 5, what is the area of the triangle?

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A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{3 π}{8}$ $(3pi)/8$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 5, what is the area of the triangle?

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Answer 1 · 2016-04-15T08:38:06

Hence area of triangle is $3.015$ $3.015$ units

Explanation:

The length of side $B$ $B$ is $5$ $5$ and angle opposite this side is angle between sides $A$ $A$ and $C$ $C$ , which is not given. But as other two angles are $\frac{3 π}{8}$ $(3pi)/8$ and $\frac{π}{12}$ $pi/12$ , this angle would be

$π - \frac{3 π}{8} - \frac{π}{12} = \frac{24 π - 9 π - 2 π}{24} = \frac{13 π}{24}$ $pi-(3pi)/8-pi/12=(24pi-9pi-2pi)/24=(13pi)/24$

Now for using sine formula for area of triangle given by $\frac{1}{2} \times a b \times sin θ$ $1/2xxabxxsintheta$ , we need one more side. Let us choose the side $C$ $C$ , which is opposite the angle $\frac{3 π}{8}$ $(3pi)/8$ .

Now using sine rule we have

$\frac{5}{sin (\frac{13 π}{24})} = \frac{C}{sin (\frac{3 π}{8})}$ $5/sin((13pi)/24)=C/sin((3pi)/8)$ or $C = 5 \times \frac{sin (\frac{3 π}{8})}{sin (\frac{13 π}{24})}$ $C=5xxsin((3pi)/8)/sin((13pi)/24)$

Hence area of triangle is $\frac{1}{2} \times 5 \times 5 \times \frac{sin (\frac{3 π}{8})}{sin (\frac{13 π}{24})} \times sin (\frac{π}{12})$ $1/2xx5xx5xxsin((3pi)/8)/sin((13pi)/24)xxsin(pi/12)$

= $\frac{25}{2} \times \frac{0.9239 \times 0.2588}{0.9914} = 3.015$ $25/2xx(0.9239xx0.2588)/0.9914=3.015$ units

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A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{3 π}{8}$ $(3pi)/8$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 5, what is the area of the triangle?

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A triangle has sides A,B, and C. If the angle between sides A and B is 3π8(3pi)/8, the angle between sides B and C is π12pi/12, and the length of B is 5, what is the area of the triangle?

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A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{3 π}{8}$ $(3pi)/8$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 5, what is the area of the triangle?