A triangle has sides A, B, and C. If the angle between sides A and B is $\frac{π}{6}$ $(pi)/6$ , the angle between sides B and C is $\frac{5 π}{12}$ $(5pi)/12$ , and the length of B is 2, 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 $\frac{π}{6}$ $(pi)/6$ , the angle between sides B and C is $\frac{5 π}{12}$ $(5pi)/12$ , and the length of B is 2, what is the area of the triangle?

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Answer 1 · 2016-01-03T07:42:39

$A r e a = 1.93184$ $Area=1.93184$ square units

Explanation:

First of all let me denote the sides with small letters a, b and c
Let me name the angle between side "a" and "b" by $∠ C$ $/_ C$ , angle between side "b" and "c" $∠ A$ $/_ A$ and angle between side "c" and "a" by $∠ B$ $/_ B$ .

Note:- the sign $∠$ $/_$ is read as "angle".
We are given with $∠ C$ $/_C$ and $∠ A$ $/_A$ . We can calculate $∠ B$ $/_B$ by using the fact that the sum of any triangles' interior angels is pi radian.
$\Rightarrow ∠ A + ∠ B + ∠ C = π$ $implies /_A+/_B+/_C=pi$
$\Rightarrow \frac{π}{6} + ∠ B + \frac{5 π}{12} = π$ $implies pi/6+/_B+(5pi)/12=pi$
$\Rightarrow ∠ B = π - \frac{7 π}{12} = \frac{5 π}{12}$ $implies/_B=pi-(7pi)/12=(5pi)/12$
$\Rightarrow ∠ B = \frac{5 π}{12}$ $implies /_B=(5pi)/12$

It is given that side $b = 2 .$ $b=2.$

Using Law of Sines
$\frac{sin ∠ B}{b} = \frac{sin ∠ C}{c}$ $(Sin/_B)/b=(sin/_C)/c$
$\Rightarrow \frac{sin (\frac{5 π}{12})}{2} = \frac{sin (\frac{5 π}{12})}{c}$ $implies (Sin((5pi)/12))/2=sin((5pi)/12)/c$
$\Rightarrow \frac{1}{2} = \frac{1}{c}$ $implies 1/2=1/c$
$\Rightarrow c = 2$ $implies c=2$

Therefore, side $c = 2$ $c=2$

Area is also given by
$A r e a = \frac{1}{2} b c sin ∠ A = \frac{1}{2} \cdot 2 \cdot 2 sin (\frac{7 π}{12}) = 2 \cdot 0.96592 = 1.93184$ $Area=1/2bcSin/_A=1/2*2*2Sin((7pi)/12)=2*0.96592=1.93184$ square units
$\Rightarrow A r e a = 1.93184$ $implies Area=1.93184$ square units

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A triangle has sides A, B, and C. If the angle between sides A and B is $\frac{π}{6}$ $(pi)/6$ , the angle between sides B and C is $\frac{5 π}{12}$ $(5pi)/12$ , and the length of B is 2, 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 $\frac{π}{6}$ $(pi)/6$ , the angle between sides B and C is $\frac{5 π}{12}$ $(5pi)/12$ , and the length of B is 2, what is the area of the triangle?