Answer
$a=20$ $;$ $b=25$ $;$ $c=22$
$\angle A\approx49.87^{\circ}$ $;$ $\angle B\approx72.89^{\circ}$ $;$ $\angle C\approx57.24^{\circ}$
Work Step by Step
$a=20$ $,$ $b=25$ $,$ $c=22$
Find angle $A$ using the formula $a^{2}=b^{2}+c^{2}-2bc\cos A$, obtained from the Law of Cosines. Substitute the known values into the formula and solve for $A$:
$20^{2}=25^{2}+22^{2}-2(25)(22)\cos A$
$(2)(25)(22)\cos A=25^{2}+22^{2}-20^{2}$
$1100\cos A=625+484-400$
$\cos A=\dfrac{625+484-400}{1100}$
$A=\cos^{-1}\Big(\dfrac{709}{1100}\Big)\approx49.87^{\circ}$
Find angle $B$ using the formula $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}$, obtained from the Law of Sines. Substitute the known values into the formula and solve for $B$:
$\dfrac{\sin49.87^{\circ}}{20}=\dfrac{\sin B}{25}$
$\sin B=\Big(\dfrac{25}{20}\Big)\sin49.87^{\circ}$
$\sin B=\Big(\dfrac{5}{4}\Big)\sin49.87^{\circ}$
$B=\sin^{-1}\Big[\Big(\dfrac{5}{4}\Big)\sin49.87^{\circ}\Big]\approx72.89^{\circ}$
Two angles are now known. Since $\angle A+\angle B+\angle C=180^{\circ}$, substitute the known angles into the formula and solve for $C$:
$\angle C=180^{\circ}-49.87^{\circ}-72.89^{\circ}\approx57.24^{\circ}$
