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 $8$ $8$ 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 · 2017-05-18T17:30:55

$A 00 \frac{π}{12} 00000 a 00 2.391$ $Acolor(white)(00)pi/12color(white)(00000)acolor(white)(00)color(red)(2.391)$

$B 00 \frac{7 π}{12} 00000 b 00 8.923$ $Bcolor(white)(00)color(black)((7pi)/12)color(white)(00000)bcolor(white)(00)8.923$

$C 00 \frac{π}{3} 000000 c 008$ $Ccolor(white)(00)pi/3color(white)(0)color(white)(00000)c color(white)(00)8$

Let's write this information in a table, where capital letters correspond to angles , lowercase to lengths :

$A 00 \frac{π}{12} 00000 a 00$ $Acolor(white)(00)pi/12color(white)(00000)acolor(white)(00)$

$B 00 \frac{π}{12} 00000 b 00$ $Bcolor(white)(00)color(white)(pi/12)color(white)(00000)bcolor(white)(00)$

$C 00 \frac{π}{3} 000000 c 008$ $Ccolor(white)(00)pi/3color(white)(0)color(white)(00000)c color(white)(00)8$

All angles in a triangle add up to $180^{o}$ $180^o$ , or $π$ $pi$

$\frac{π}{12} + \frac{π}{3}$ $pi/12+pi/3$ only equals $\frac{5 π}{12}$ $(5pi)/12$ .

$\frac{12 π}{12} - \frac{5 π}{12} = \frac{7 π}{12}$ $(12pi)/12-(5pi)/12=(7pi)/12$

That's the remaining angle:

$A 00 \frac{π}{12} 00000 a 00$ $Acolor(white)(00)pi/12color(white)(00000)acolor(white)(00)$

$B 00 \frac{7 π}{12} 00000 b 00$ $Bcolor(white)(00)color(black)((7pi)/12)color(white)(00000)bcolor(white)(00)$

$C 00 \frac{π}{3} 000000 c 008$ $Ccolor(white)(00)pi/3color(white)(0)color(white)(00000)c color(white)(00)8$

We can use law of sines to find the other lengths:

$\frac{sin (\frac{π}{3})}{8} = \frac{sin (\frac{π}{12})}{a}$ $(sin(pi/3))/8=(sin(pi/12))/a$

$a \approx 2.391$ $a~~2.391$

Just, for fun, let's also find length b

$\frac{sin (\frac{π}{3})}{8} = \frac{sin (\frac{7 π}{12})}{b}$ $(sin(pi/3))/8=(sin((7pi)/12))/b$

$b \approx 8.923$ $b~~8.923$

A triangle has sides A, B, and C. The angle between sides A and B is π3pi/3. If side C has a length of 88 and the angle between sides B and C is π12pi/12, what is the length of side A?