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

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Answer 1 · 2016-01-15T11:55:06

$area = 1.125 {units}^{2} \to 1 \frac{1}{8} {units}^{2}$ $color(green)("area " = 1.125 " units"^2 -> 1 1/8 " units"^2)$

Explanation:

$Assumption:$ $color(blue)("Assumption: ")$

As $π$ $pi$ is used in the angular measure it is assumed that the unit is radians. (Not stated)

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Tony B

$Method Plane$ $color(blue)("Method Plane")$

Determine $∠ c b a$ $/_cba$

Using Sine Rule and $∠ c b a$ $/_cba$ determine length of side A
Determine h using $h = A sin (\frac{5 π}{12})$ $h=Asin((5pi)/12)$
Determine area $h \times \frac{B}{2}$ $hxxB/2$

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$To determine ∠ c b a$ $color(blue)("To determine" /_cba)$

Sum internal angles of a triangle is $180^{0} = π radians$ $180^0 = pi" radians"$

$\Rightarrow ∠ c b a = π - \frac{5 π}{12} - \frac{π}{12}$ $=>/_cba= pi-(5pi)/12-pi/12$
$∠ c b a = \frac{π}{2} \to 90^{o}$ $color(brown)(/_cba = pi/2 -> 90^o)$

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$Determine length of A$ $color(blue)("Determine length of A")$

Using $\frac{B}{sin (b)} = \frac{A}{sin (a)}$ $B/(sin(b))=A/sin(a)$

$\Rightarrow \frac{3}{sin (\frac{π}{2})} = \frac{A}{sin (\frac{π}{12})}$ $=> 3/(sin(pi/2)) =A/sin(pi/12)$

$\Rightarrow A = \frac{3 \times sin (\frac{π}{12})}{sin (\frac{π}{2})}$ $=> A= (3xxsin(pi/12))/(sin(pi/2))$

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

$\Rightarrow A = 3 \times sin (\frac{π}{12})$ $color(blue)(=> A = 3xxsin(pi/12))$

$This is an exact value so keep it in this form for now to reduce error$ $color(brown)("This is an exact value so keep it in this form for now to reduce error")$ $on final calculation.$ $color(brown)("on final calculation.")$
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$To determine h$ $color(blue)("To determine h")$

$h = A sin (\frac{5 π}{12})$ $h=Asin((5pi)/12)$

$\Rightarrow h = 3 \times sin (\frac{π}{12}) \times sin (\frac{5 π}{12})$ $=>h=3xxsin(pi/12)xxsin((5pi)/12)$

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$To determine area$ $color(blue)("To determine area")$

$area = \frac{B}{2} \times h$ $"area "= B/2xxh$

$area = \frac{3}{2} \times 3 \times sin (\frac{π}{12}) \times sin (\frac{5 π}{12})$ $"area "= 3/2 xx3xx sin(pi/12)xxsin((5pi)/12)$

but $sin (\frac{π}{12}) \times sin (\frac{5 π}{12}) = \frac{1}{4}$ $sin(pi/12)xxsin((5pi)/12)=1/4$

$area = \frac{3}{2} \times 3 \times \frac{1}{4}$ $"area "= 3/2 xx3xx1/4$

$area = 1.125 {units}^{2} \to 1 \frac{1}{8} {units}^{2}$ $color(green)("area " = 1.125 " units"^2 -> 1 1/8 " units"^2)$

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{5 π}{12}$ $(5pi)/12$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 3, 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 5π12(5pi)/12 and the angle between sides B and C is π12pi/12. If side B has a length of 3, 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 $\frac{5 π}{12}$ $(5pi)/12$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ . If side B has a length of 3, what is the area of the triangle?