A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 5, respectively. The angle between A and C is $\frac{π}{3}$ $(pi)/3$ and the angle between B and C is $\frac{π}{4}$ $(pi)/4$ . What is the area of the triangle?

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A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 5, respectively. The angle between A and C is $\frac{π}{3}$ $(pi)/3$ and the angle between B and C is $\frac{π}{4}$ $(pi)/4$ . What is the area of the triangle?

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Answer 1 · 2016-01-19T09:09:42

$A \approx 6.54$ $A~~6.54$

Explanation:

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The area of a triangle is given by $\frac{1}{2} \cdot B a s e \cdot h e i g h t$ $1/2*Base*height$
The height $h$ $h$ in this example can be found from
$sin (\frac{π}{3}) = \frac{h}{A} = \frac{h}{3}$ $sin(pi/3) =h/A = h/3$
$h = 3 sin (\frac{π}{3})$ $h = 3sin(pi/3)$

The base $C = x + y$ $C = x +y$ and $x$ $x$ and $y$ $y$ can also be found from trigonometric functions.
$\frac{x}{A} = cos (\frac{π}{3})$ $x/A = cos(pi/3)$ so $x = 3 cos (\frac{π}{3})$ $x = 3cos(pi/3)$
$\frac{y}{B} = cos (\frac{π}{4})$ $y/B = cos(pi/4)$ so $y = 5 cos (\frac{π}{4})$ $y = 5cos(pi/4)$

Area $A = \frac{1}{2} \cdot (3 cos (\frac{π}{3}) + 5 cos (\frac{π}{4})) \cdot 3 sin (\frac{π}{3})$ $A = 1/2*(3cos(pi/3)+5cos(pi/4))*3sin(pi/3)$

$A = \frac{1}{2} \cdot (3 \cdot 0.5 + 5 \cdot 0.7071) \cdot 3 \cdot 0.866$ $A =1/2*(3*0.5 +5*0.7071)*3*0.866$
$A \approx 6.54$ $A~~6.54$

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A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 5, respectively. The angle between A and C is $\frac{π}{3}$ $(pi)/3$ and the angle between B and C is $\frac{π}{4}$ $(pi)/4$ . What is the area of the triangle?

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A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 5, respectively. The angle between A and C is (pi)/3π3(pi)/3 and the angle between B and C is (pi)/4π4 (pi)/4. What is the area of the triangle?

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A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 5, respectively. The angle between A and C is $\frac{π}{3}$ $(pi)/3$ and the angle between B and C is $\frac{π}{4}$ $(pi)/4$ . What is the area of the triangle?