A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{7 π}{12}$ $(7pi)/12$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , 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 $\frac{7 π}{12}$ $(7pi)/12$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 2, what is the area of the triangle?

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Answer 1 · 2016-01-09T18:48:34

$A r e a = 0.5773$ $Area=0.5773$ square units

Explanation:

First of all let me denote the sides with small letters $a$ $a$ , $b$ $b$ and $c$ $c$ .
Let me name the angle between side $a$ $a$ and $b$ $b$ by $∠ C$ $/_ C$ , angle between side $b$ $b$ and $c$ $c$ by $∠ A$ $/_ A$ and angle between side $c$ $c$ and $a$ $a$ 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 $π$ $pi$ radian.
$\Rightarrow ∠ A + ∠ B + ∠ C = π$ $implies /_A+/_B+/_C=pi$
$\Rightarrow \frac{π}{12} + ∠ B + \frac{7 π}{12} = π$ $implies pi/12+/_B+(7pi)/12=pi$
$\Rightarrow ∠ B = π - (\frac{π}{12} + \frac{7 π}{12}) = π - \frac{8 π}{12} = π - \frac{2 π}{3} = \frac{π}{3}$ $implies/_B=pi-(pi/12+(7pi)/12)=pi-(8pi)/12=pi-(2pi)/3=pi/3$
$\Rightarrow ∠ B = \frac{π}{3}$ $implies /_B=pi/3$

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

Using Law of Sines

$\frac{sin ∠ B}{b} = \frac{sin ∠ C}{c}$ $(Sin/_B)/b=(sin/_C)/c$

$\Rightarrow \frac{sin (\frac{π}{3})}{2} = \frac{sin (\frac{7 π}{12})}{c}$ $implies (Sin((pi)/3))/2=sin((7pi)/12)/c$

$\Rightarrow \frac{0.86602}{2} = \frac{0.96592}{c}$ $implies (0.86602)/2=(0.96592)/c$

$\Rightarrow 0.43301 = \frac{0.96592}{c}$ $implies 0.43301=0.96592/c$

$\Rightarrow c = \frac{0.96592}{0.43301}$ $implies c=0.96592/0.43301$

$\Rightarrow c = 2.2307$ $implies c=2.2307$

Therefore, side $c = 2.2307$ $c=2.2307$

Area is also given by
$A r e a = \frac{1}{2} b c sin ∠ A$ $Area=1/2bcSin/_A$

$\Rightarrow A r e a = \frac{1}{2} \cdot 2 \cdot 2.2307 sin (\frac{π}{12}) = 2.2307 \cdot 0.2588 = 0.5773$ $implies Area=1/2*2*2.2307Sin((pi)/12)=2.2307*0.2588=0.5773$ square units
$\Rightarrow A r e a = 0.5773$ $implies Area=0.5773$ square units

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A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{7 π}{12}$ $(7pi)/12$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 2, what is the area of the triangle?

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Explanation:

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A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{7 π}{12}$ $(7pi)/12$ , the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , and the length of B is 2, what is the area of the triangle?