Answer
The angles of the triangle are as follows:
$A = 25.2^{\circ}, B = 17.3^{\circ},$ and $C = 137.5^{\circ}$
The lengths of the sides are as follows:
$a = 6.92~yd, b = 4.82~yd,$ and $c = 10.98~yd$
Work Step by Step
$A = 25.2^{\circ}$
$a = 6.92~yd$
$b = 4.82~yd$
We can use the law of sines to find angle $B$:
$\frac{a}{sin~A} = \frac{b}{sin~B}$
$sin~B = \frac{b~sin~A}{a}$
$B = arcsin(\frac{b~sin~A}{a})$
$B = arcsin(\frac{4.82~sin~25.2}{6.92})$
$B = arcsin(0.29657)$
$B = 17.3^{\circ}$
We can find angle $C$:
$A+B+C = 180^{\circ}$
$C = 180^{\circ}-A-B$
$C = 180^{\circ}-25.2^{\circ}-17.3^{\circ}$
$C = 137.5^{\circ}$
We can use the law of sines to find the length of side $c$:
$\frac{c}{sin~C} = \frac{a}{sin~A}$
$c = \frac{a~sin~C}{sin~A}$
$c = \frac{(6.92~yd)~sin~(137.5^{\circ})}{sin~25.2^{\circ}}$
$c = 10.98~yd$
Note that another possible value for angle $B$ is $180^{\circ}-17.3^{\circ} = 162.7^{\circ}$
However, then the sum of angle A and angle B is greater than $180^{\circ}$, so this value for angle B is not acceptable.
