A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{5 π}{12}$ $(5pi)/12$ , the angle between sides B and C is $\frac{π}{6}$ $pi/6$ , and the length of B is 15, 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 $\frac{5 π}{12}$ $(5pi)/12$ , the angle between sides B and C is $\frac{π}{6}$ $pi/6$ , and the length of B is 15, what is the area of the triangle?

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Answer 1 · 2017-06-27T06:16:35

The area of the triangle is $= 56.25 u^{2}$ $=56.25u^2$

Explanation:

The angle between $A$ $A$ and $B$ $B$ is

$= π - (\frac{5}{12} π + \frac{2}{12} π)$ $=pi-(5/12pi+2/12pi)$

$= π - (\frac{7}{12} π)$ $=pi-(7/12pi)$

$= \frac{5}{12} π$ $=5/12pi$

So, the triangle is isoceles

The side $B$ $B$ is $= 15$ $=15$

Therefore, the side $C$ $C$ is $= 15$ $=15$

$sin (\frac{1}{6} π) = \frac{1}{2}$ $sin(1/6pi)=1/2$

The area of the triangle is

$A = \frac{1}{2} \cdot 15 \cdot 15 sin (\frac{1}{6} π)$ $A=1/2*15*15sin(1/6pi)$

$= \frac{225}{4} = 56.25$ $=225/4=56.25$

Answer 2 · 2017-06-27T06:30:40

$a r e a = 56.25$ $area = 56.25$ units squared

Explanation:

Here's all the info we know (I found that last angle by subtracting all the other angles from $2 π)$ $2pi)$ :
enter image source here

So, here's what we need to know for the area:
enter image source here

Once we find these values, we'll be able to use the formula $a r e a = \frac{1}{2} (b \times h)$ $area=1/2(b xx h)$

Let's work on finding the height, $h$ $h$ .

To do that, we just need to use $sin (\frac{π}{6}) = \frac{h}{15}$ $sin(pi/6)=h/15$ , or $7.5 = h$ $7.5=h$

Now we know the height, all that's left is to find the base, or $c$ $c$

First, let's put all our info in a table:

length $0000$ $color(white)(0000)$ angle
$A = ? 0000 A = \frac{π}{6}$ $A = ? color(white)(0000) A= pi/6$
$B = 150000 B = \frac{17 π}{12}$ $B = 15 color(white)(0000) B = (17pi)/12$
$C = ? 0000 A = \frac{5 π}{12}$ $C = ? color(white)(0000) A= (5pi)/12$

Let's use law of sines

$\frac{sin (\frac{5 π}{12})}{C} = \frac{sin (\frac{17 π}{12})}{15}$ $(sin((5pi)/12))/(C) = (sin((17pi)/12))/(15)$

$c = - 15$ $c = -15$ , but because this problem is dealing in distances, we cannot have a negative length, so $c = 15$ $c=15$

Now let's use our formula: $a r e a = \frac{1}{2} (b \times h)$ $area=1/2(b xx h)$

$a r e a = \frac{15 \times 7.5}{2}$ $area = (15 xx 7.5)/2$

$a r e a = 56.25$ $area = 56.25$ units squared

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A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{5 π}{12}$ $(5pi)/12$ , the angle between sides B and C is $\frac{π}{6}$ $pi/6$ , and the length of B is 15, 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 5π12(5pi)/12, the angle between sides B and C is π6pi/6, and the length of B is 15, what is the area of the triangle?

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Explanation:

Explanation:

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A triangle has sides A,B, and C. If the angle between sides A and B is $\frac{5 π}{12}$ $(5pi)/12$ , the angle between sides B and C is $\frac{π}{6}$ $pi/6$ , and the length of B is 15, what is the area of the triangle?