A triangle has sides A, B, and C. The angle between sides A and B is $\frac{5 π}{6}$ $(5pi)/6$ . If side C has a length of $36$ $36$ 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 · 2016-03-10T13:05:25

$\Rightarrow A = B \approx 18.635$ $=> A = B ~~ 18.635$ to 3 decimal places

A Diagram always helps!

Tony B

Known: The sum of the internal angles is $π$ $pi$ radians ( $180^{o})$ $180^o)$

Thus the angle between A and C is $π - \frac{π}{12} - \frac{5 π}{6} = \frac{π}{12}$ $pi-pi/12-(5pi)/6 = pi/12$

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$Using the sine rule$ $color(blue)("Using the sine rule")$

$\frac{C}{sin (∠ A B)} = \frac{B}{sin (∠ A C)} = \frac{A}{sin (∠ B C)}$ $C/(sin(/_AB)) =B/(sin(/_AC))=A/(sin(/_BC))$

Also as $∠ A C = ∠ B C$ $/_AC = /_BC$ then length $A = B$ $A=B$

$\frac{C}{sin (∠ A B)} = \frac{36}{sin (\frac{5 π}{6})} = \frac{A}{sin (\frac{π}{12})}$ $C/(sin(/_AB)) = 36/(sin((5pi)/6))= A/(sin( pi/12))$

$\Rightarrow A = \frac{sin (\frac{π}{12})}{sin (\frac{5 π}{6})} \times 36$ $=>A = (sin(pi/12))/(sin((5pi)/6))xx36$

$\Rightarrow A = B \approx 18.635$ $=> A = B ~~ 18.635$ to 3 decimal places

A triangle has sides A, B, and C. The angle between sides A and B is 5π6(5pi)/6. If side C has a length of 3636 and the angle between sides B and C is π12pi/12, what are the lengths of sides A and B?