A triangle has sides A, B, and C. The angle between sides A and B is $\frac{3 π}{4}$ $(3pi)/4$ . If side C has a length of $9$ $9$ and the angle between sides B and C is $\frac{π}{12}$ $pi/12$ , what are the lengths of sides A and B?

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Answer 1 · 2016-07-06T11:33:18

$A = \frac{9}{2} (\sqrt{3} - 1)$ $A=9/2(sqrt(3)-1)$ ; $B = \frac{9}{2} \sqrt{2}$ $B=9/2sqrt(2)$

You can use the equation:

$\frac{A}{sin ˆ B C} = \frac{C}{sin ˆ A B}$ $A/sin hat(BC)=C/sin hat(AB)$

that, in this case, is:

$\frac{A}{sin (\frac{π}{12})} = \frac{9}{sin (3 \frac{π}{4})}$ $A/sin(pi/12)=9/sin(3pi/4)$

$A = 9 \frac{\frac{\sqrt{6} - \sqrt{2}}{4}}{\frac{\sqrt{2}}{2}}$ $A=9((sqrt(6)-sqrt(2))/4)/(sqrt(2)/2)$

$A = 9 (\frac{\sqrt{6} - \sqrt{2}}{4}) \cdot \frac{2}{\sqrt{2}}$ $A=9((sqrt(6)-sqrt(2))/4)*2/sqrt(2)$

$A = 9 (\frac{\sqrt{6} - \sqrt{2}}{2 \sqrt{2}})$ $A=9((sqrt(6)-sqrt(2))/(2sqrt(2)))$

$A = 9 ⎛ ⎜ ⎝ \frac{(\sqrt{6} - \sqrt{2}) \cdot \sqrt{2}}{2 \sqrt{2} \sqrt{2}} ⎞ ⎟ ⎠$ $A=9(((sqrt(6)-sqrt(2))*sqrt(2))/(2sqrt(2)sqrt(2)))$

$A = 9 \frac{\sqrt{12} - 2}{4}$ $A=9(sqrt(12)-2)/(4)$

$A = \frac{9}{2} (\sqrt{3} - 1)$ $A=9/2(sqrt(3)-1)$

Now let's calculate $ˆ A C$ $hat(AC)$ and B

$ˆ A C = π - (\frac{π}{12} + 3 \frac{π}{4}) = \frac{π}{6}$ $hat(AC)=pi-(pi/12+3pi/4)=pi/6$

$\frac{B}{sin ˆ A C} = \frac{C}{sin (ˆ A B)}$ $B/sin hat(AC)=C/sin(hat (AB))$

$\frac{B}{sin (\frac{π}{6})} = \frac{9}{sin (3 \frac{π}{4})} ⎞ ⎟ ⎠$ $B/sin (pi/6)=9/sin(3pi/4))$

$B = \frac{9 \cdot \frac{1}{2}}{\frac{\sqrt{2}}{2}} = \frac{9}{\sqrt{2}} = \frac{9}{2} \sqrt{2}$ $B=(9*1/2)/(sqrt(2)/2)=9/sqrt(2)=9/2sqrt(2)$

A triangle has sides A, B, and C. The angle between sides A and B is 3π4(3pi)/4. If side C has a length of 99 and the angle between sides B and C is π12pi/12, what are the lengths of sides A and B?