A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{3}$ $(pi)/3$ and the angle between sides B and C is $\frac{π}{6}$ $pi/6$ . If side B has a length of 13, 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 $\frac{π}{3}$ $(pi)/3$ and the angle between sides B and C is $\frac{π}{6}$ $pi/6$ . If side B has a length of 13, what is the area of the triangle?

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Answer 1 · 2016-12-26T06:29:33

$A_{△} = \frac{169 \sqrt{3}}{8} \approx 36.59$ $A_triangle=(169sqrt 3)/8 approx 36.59$ .

Explanation:

Our goal will be to use $A_{△} = \frac{1}{2} a b sin C$ $A_triangle = 1/2 a b sin C$ . We know $b = 13$ $b=13$ and $∠ C = \frac{π}{3}$ $angle C = pi/3$ , so we need to find $a$ $a$ .

Step 1: Find the value of $∠ B$ $angle B$ .

Using the fact that the sum of all 3 angles in a triangle is $π$ $pi$ , we get

$∠ A + ∠ B + ∠ C = π$ $angle A + angle B + angle C = pi$
$\frac{π}{6} + ∠ B + \frac{π}{3} = π$ $pi/6" "+ angle B + pi / 3" "= pi$
$∠ B = \frac{π}{2}$ $" "angle B " "= pi/2$

So $∠ B = \frac{π}{2}$ $angle B = pi/2$ .

Step 2: Find the length of $a$ $a$ .

We now use the sine law for triangles to get

$\frac{a}{sin A} = \frac{b}{sin B}$ $a/sinA=b/sinB$

$\frac{a}{sin (\frac{π}{6})} = \frac{13}{sin (\frac{π}{2})}$ $a/sin(pi/6)=13/sin(pi/2)$

$a = \frac{13 sin (\frac{π}{6})}{sin (\frac{π}{2})}$ $" "a" "=(13sin(pi/6))/sin(pi/2)$

$a = \frac{13 (\frac{1}{2})}{1} = \frac{13}{2}$ $" "a" "=(13(1/2))/(1)=13/2$

So $a = \frac{13}{2}$ $a=13/2$ .

Step 3: Find the area of the triangle.

We can now use the following formula for a triangle's area:

$A_{△} = \frac{1}{2} a b sin C$ $A_triangle=1/2 a b sin C$

$A_{△} = \frac{1}{2} \cdot \frac{13}{2} \cdot 13 \cdot sin (\frac{π}{3})$ $A_triangle=1/2 * 13/2 * 13 * sin (pi/3)$

$A_{△} = \frac{169}{4} \cdot \frac{\sqrt{3}}{2}$ $A_triangle=169/4 * sqrt 3 / 2$

$A_{△} = \frac{169 \sqrt{3}}{8} \approx 36.59$ $A_triangle=(169sqrt 3)/8" "approx 36.59$ .

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{π}{3}$ $(pi)/3$ and the angle between sides B and C is $\frac{π}{6}$ $pi/6$ . If side B has a length of 13, 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 $\frac{π}{3}$ $(pi)/3$ and the angle between sides B and C is $\frac{π}{6}$ $pi/6$ . If side B has a length of 13, what is the area of the triangle?