Answer
The angles of the triangle are as follows:
$A = 24.1^{\circ}, B = 30.8^{\circ},$ and $C = 125.1^{\circ}$
The lengths of the sides are as follows:
$a = 4.0~ft, b = 5.0~ft,$ and $c = 8.0~ft$
Work Step by Step
We can use the law of cosines to find $B$:
$b^2 = a^2+c^2-2ac~cos~B$
$2ac~cos~B = a^2+c^2-b^2$
$cos~B = \frac{a^2+c^2-b^2}{2ac}$
$B = arccos(\frac{a^2+c^2-b^2}{2ac})$
$B = arccos(\frac{4.0^2+8.0^2-5.0^2}{(2)(4.0)(8.0)})$
$B = arccos(0.859375)$
$B = 30.8^{\circ}$
We can use the law of cosines to find $C$:
$c^2 = a^2+b^2-2ab~cos~C$
$2ab~cos~C = a^2+b^2-c^2$
$cos~C = \frac{a^2+b^2-c^2}{2ab}$
$C = arccos(\frac{a^2+b^2-c^2}{2ab})$
$C = arccos(\frac{4.0^2+5.0^2-8.0^2}{(2)(4.0)(5.0)})$
$C = arccos(-0.575)$
$C = 125.1^{\circ}$
We can find angle $A$:
$A+B+C = 180^{\circ}$
$A = 180^{\circ}-B-C$
$A = 180^{\circ}-30.8^{\circ}-125.1^{\circ}$
$A = 24.1^{\circ}$
