Answer
$a\approx48.736$ $;$ $b=25$ $;$ $c=30$
$\angle A=124.526^{\circ}$ $;$ $\angle B=25^{\circ}$ $;$ $\angle C\approx30.474^{\circ}$
Work Step by Step
$b=25$ $,$ $c=30$ $,$ $\angle B=25^{\circ}$
The Law of Sines is $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$
Find angle $C$ by using the formula $\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$, obtained from the Law of Sines. Substitute the known values and solve for $\angle C$:
$\dfrac{\sin25^{\circ}}{25}=\dfrac{\sin C}{30}$
$\sin C=\Big(\dfrac{30}{25}\Big)\sin25^{\circ}$
$C=\sin^{-1}\Big[\Big(\dfrac{30}{25}\Big)\sin25^{\circ}\Big]\approx30.474^{\circ}$
Two angles are now known. Since $\angle A+\angle B+\angle C=180^{\circ}$. Substitute the known angles into the formula and solve for $\angle A$:
$A=180^{\circ}-25^{\circ}-30.474^{\circ}\approx124.526^{\circ}$
Find the side $a$ by using the formula $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}$, obtained from the Law of Sines. Substitute the known values and solve for $a$:
$\dfrac{\sin124.526^{\circ}}{a}=\dfrac{\sin25^{\circ}}{25}$
$\dfrac{a}{\sin124.526^{\circ}}=\dfrac{25}{\sin25^{\circ}}$
$a=\Big(\dfrac{\sin124.526^{\circ}}{\sin25^{\circ}}\Big)25\approx48.736$
