A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 2, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{π}{8}$ $(pi)/8$ . 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 2, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{π}{8}$ $(pi)/8$ . What is the area of the triangle?

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Answer 1 · 2016-05-20T06:39:21

If $A = 3$ $A=3$ then area is $7.57$ $7.57$ and if $B = 2$ $B=2$ area is $0.746$ $0.746$

Explanation:

The third angle opposite sides $A$ $A$ and $B$ $B$ is

$π - \frac{11 π}{24} - \frac{π}{8} = (24 - 11 - 3) \frac{π}{24} = \frac{10 π}{24} = \frac{5 π}{12}$ $pi-(11pi)/24-pi/8=(24-11-3)pi/24=(10pi)/24=(5pi)/12$ and it has side $C$ $C$ opposite it.

As side $A = 3$ $A=3$ has angle opposite it $\frac{π}{8}$ $pi/8$ and $C$ $C$ has opposite to it angle $\frac{5 π}{12}$ $(5pi)/12$ . Now, using sine formula, we get

$\frac{3}{sin (\frac{π}{8})} = \frac{C}{sin (\frac{5 π}{12})}$ $3/sin(pi/8)=C/sin((5pi)/12)$ or

$C = 3 \times \frac{sin (\frac{5 π}{12})}{sin (\frac{π}{8})} = 3 \times \frac{0.9659}{0.3827} = 7.57$ $C=3xxsin((5pi)/12)/sin(pi/8)=3xx0.9659/0.3827=7.57$

Hence area of triangle is $\frac{1}{2} \times 3 \times 7.57 \times sin (\frac{11 π}{24})$ $1/2xx3xx7.57xxsin((11pi)/24)$

= $\frac{1}{2} \times 3 \times 7.57 \times 0.9914 = 11.26$ $1/2xx3xx7.57xx0.9914=11.26$

We have not used $B = 2$ $B=2$ and as angle opposite it is $(\frac{11 π}{24})$ $((11pi)/24)$ , using sine formula

$\frac{2}{sin (\frac{11 π}{24})} = \frac{C}{sin (\frac{5 π}{12})}$ $2/sin((11pi)/24)=C/sin((5pi)/12)$

or $C = 2 \times \frac{sin (\frac{5 π}{12})}{sin (\frac{11 π}{24})} = 2 \times \frac{0.9659}{0.9914} = 1.95$ $C=2xxsin((5pi)/12)/sin((11pi)/24)=2xx0.9659/0.9914=1.95$

and area of triangle is $\frac{1}{2} \times 2 \times 1.95 \times sin (\frac{π}{8}) = 1.95 \times 0.3827 = 0.746$ $1/2xx2xx1.95xxsin(pi/8)=1.95xx0.3827=0.746$

Why this dichotomy? The fact is that we need either (a) one side and both angles on it; or (b) two sides and included angle and (iii) three sides of a triangle to identify a triangle and find area or other sides and angles of a triangle. However here we have been given four parameters and they give two different results depending on whether we take side $A = 3$ $A=3$ or $B = 2$ $B=2$ into consideration. In short, given three angles (third is derivable from other two), the two sides are not compatible and in fact refer to two different triangles.

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A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 2, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{π}{8}$ $(pi)/8$ . 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 2, respectively. The angle between A and C is 11π24(11pi)/24 and the angle between B and C is π8 (pi)/8. 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 2, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{π}{8}$ $(pi)/8$ . What is the area of the triangle?