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

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

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Answer 1 · 2016-05-19T17:34:01

$2 \sqrt{6} (\sqrt{3} - 1)$ $2 sqrt(6) (sqrt(3)-1)$

Explanation:

Assuming angles opposite to sides $A, B$ $A, B$ and $C$ $C$ are $∠ A, ∠ B and ∠ C$ $/_A, /_B and /_C$ , respectively.

Then
$∠ C = \frac{π}{3} and ∠ A = \frac{π}{12}$ $/_C = pi/3 and /_A = pi/12$

Using Sine Rule

$\frac{sin ∠ A}{A} = \frac{sin ∠ B}{B} = \frac{sin ∠ C}{C}$ $(Sin/_A)/A = (Sin/_B)/B = (Sin/_C)/C$

we have,
$\frac{sin ∠ A}{A} = \frac{sin ∠ C}{C}$ $(Sin/_A)/A = (Sin/_C)/C$
$\frac{sin (\frac{π}{12})}{A} = \frac{sin (\frac{π}{3})}{12}$ $(Sin(pi/12))/A = (Sin(pi/3))/12$
$A = \frac{\sqrt{3} - 1}{2 \sqrt{2}} \cdot 12 \cdot \frac{1}{\frac{\sqrt{3}}{2}}$ $A=(sqrt(3)-1)/(2 sqrt(2))*12*1/(sqrt3/2)$
$or, A = 2 \sqrt{6} (\sqrt{3} - 1)$ $or, A=2 sqrt(6) (sqrt(3)-1)$
$or, A \approx 3.586$ $or, A~~3.586$

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

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