A triangle has sides A, B, and C. The angle between sides A and B is $\frac{7 π}{12}$ $(7pi)/12$ . If side C has a length of $1$ $1$ 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 $\frac{7 π}{12}$ $(7pi)/12$ . If side C has a length of $1$ $1$ 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 · 2016-01-25T15:59:35

The length of side $A$ $A$ is $2 - \sqrt{3}$ $2 - sqrt(3)$ .

Explanation:

You can use the law of sines.

The angle between sides $A$ $A$ and $B$ $B$ is the angle opposing the side $C$ $C$ . Let's call this angle $γ$ $gamma$ .

Similarly, the angle between sides $B$ $B$ and $C$ $C$ is the one opposing the side $A$ $A$ . Let's call this angle $α$ $alpha$ .

Even though we don't need it for this particular task, let's also call the last remaining angle $β$ $beta$ , this one is opposing the side $B$ $B$ .

According to the law of sines, the following relation between the sides and the opposite angles exists:

$\frac{A}{sin α} = \frac{B}{sin β} = \frac{C}{sin γ}$ $A / sin alpha = B / sin beta = C / sin gamma$

In our case, we only need to look at $A$ $A$ , $C$ $C$ , $α$ $alpha$ and $γ$ $gamma$ :

$\frac{A}{sin α} = \frac{C}{sin γ}$ $A / sin alpha = C / sin gamma$

$\frac{A}{sin (\frac{π}{12})} = \frac{1}{sin (\frac{7 π}{12})}$ $A / sin (pi / 12) = 1 / sin ((7 pi)/12)$

$\Leftrightarrow A = \frac{sin (\frac{π}{12})}{sin (\frac{7 π}{12})}$ $<=> A = sin (pi / 12) / sin ((7pi) / 12)$

So, the only thing left to do is compute $sin (\frac{π}{12})$ $sin (pi/12)$ and $sin (\frac{7 π}{12})$ $sin((7pi)/12)$ .

Let me show you how to do this without the calculator but with some basic knowledge about $sin$ $sin$ and $cos$ $cos$ and using the identity

$sin (a - b) = sin (a) cos (b) - cos (a) sin (b)$ $sin (a-b) = sin(a)cos(b) - cos(a)sin(b)$ .

You need to express $\frac{π}{12}$ $pi/12$ as a sum or difference of simpler values:

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

$= sin (\frac{π}{3}) cos (\frac{π}{4}) - cos (\frac{π}{3}) sin (\frac{π}{4})$ $= sin(pi/3) cos(pi/4) - cos(pi/3)sin(pi/4)$

$= \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} - \frac{1}{2} \cdot \frac{1}{\sqrt{2}}$ $= sqrt(3)/2 * 1/sqrt(2) - 1 / 2 * 1 / sqrt(2)$

$= \frac{1}{2 \sqrt{2}} (\sqrt{3} - 1)$ $= 1/(2 sqrt(2)) (sqrt(3) - 1)$

Similarly,

$sin (\frac{7 π}{12}) = sin (\frac{π}{3} + \frac{π}{4})$ $sin((7 pi)/12) = sin(pi/3 + pi/4)$

$= sin (\frac{π}{3}) cos (\frac{π}{4}) + cos (\frac{π}{3}) sin (\frac{π}{4})$ $= sin(pi/3) cos(pi/4) + cos(pi/3) sin(pi/4)$

$= \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ $= sqrt(3) / 2 * sqrt(2) / 2 + 1 / 2 * sqrt(2) / 2$

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

Thus, the length of the side $A$ $A$ is

$A = \frac{sin (\frac{π}{12})}{sin (\frac{7 π}{12})}$ $A = sin (pi / 12) / sin ((7pi) / 12)$

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

$= \frac{\sqrt{3} - 1}{\sqrt{3} + 1}$ $= (sqrt(3) - 1) / (sqrt(3) + 1)$

$= \frac{(\sqrt{3} - 1) (\sqrt{3} - 1)}{(\sqrt{3} + 1) (\sqrt{3} - 1)}$ $= ((sqrt(3) - 1) color(blue)((sqrt(3)-1))) / ((sqrt(3) + 1) color(blue)((sqrt(3)-1)))$

$= \frac{{(\sqrt{3} - 1)}^{2}}{{(\sqrt{3})}^{2} - 1^{2}}$ $= (sqrt(3) -1)^2 / ((sqrt(3))^2 - 1^2 )$

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

$= 2 - \sqrt{3}$ $= 2 - sqrt(3)$

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{7 π}{12}$ $(7pi)/12$ . If side C has a length of $1$ $1$ 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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Explanation:

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A triangle has sides A, B, and C. The angle between sides A and B is $\frac{7 π}{12}$ $(7pi)/12$ . If side C has a length of $1$ $1$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what is the length of side A?