A triangle has sides A, B, and C. Sides A and B have lengths of 9 and 3, respectively. The angle between A and C is $(pi)/3$ and the angle between B and C is $(pi)/4$ . What is the area of the triangle?

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Answer 1 · 2015-12-27T07:23:26

$frac{27(1+sqrt(3))}{4sqrt{2}}$

Since the interior angles of a triangle add up to $pi$ , the angle between $A$ and $B$ , $theta$ , is

$theta = pi - pi/4 - pi/3 = frac{5pi}{12}$ .

Area of triangle is calculated using

$1/2 xx A xx B xx sin(theta)$

$= 1/2 xx 9 xx 3 xx sin(frac{5pi}{12})$

$= frac{27(1+sqrt(3))}{4sqrt{2}}$

Note:

$sin(frac{5pi}{12}) = sin(pi - pi/4 - pi/3)$

$= sin(pi/4 + pi/3)$

$= sin(pi/4)cos(pi/3) + cos(pi/4)sin(pi/3)$

$= (1/sqrt{2})(1/2) + (1/sqrt{2})(sqrt{3}/2)$

$= frac{1 + sqrt{3}}{2sqrt{2}}$

A triangle has sides A, B, and C. Sides A and B have lengths of 9 and 3, respectively. The angle between A and C is (pi)/3(pi)/3 and the angle between B and C is (pi)/4 (pi)/4. What is the area of the triangle?