Answer
$a=26$ $;$ $b\approx30.871$ $;$ $c=15$
$\angle A\approx57.176^{\circ}$ $;$ $\angle B\approx93.824^{\circ}$ $;$ $\angle C=29^{\circ}$
Work Step by Step
$a=26$ $,$ $c=15$ $,$ $\angle C=29^{\circ}$
The Law of Sines is $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$
Find angle $A$ using the formula $\dfrac{\sin A}{a}=\dfrac{\sin C}{c}$, obtained from the Law of Sines. Substitute the known values and solve for $A$:
$\dfrac{\sin A}{26}=\dfrac{\sin29^{\circ}}{15}$
$\sin A=\Big(\dfrac{26}{15}\Big)\sin29^{\circ}$
$A=\sin^{-1}\Big[\Big(\dfrac{26}{15}\Big)\sin29^{\circ}\Big]\approx57.176^{\circ}$
Two angles are now known. Since $\angle A+\angle B+\angle C=180^{\circ}$, substitute the known angles into the formula and solve for $\angle B$:
$B=180^{\circ}-29^{\circ}-57.176^{\circ}\approx93.824^{\circ}$
Find the side $b$ 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 $b$:
$\dfrac{\sin93.824^{\circ}}{b}=\dfrac{\sin29^{\circ}}{15}$
$\dfrac{b}{\sin93.824^{\circ}}=\dfrac{15}{\sin29^{\circ}}$
$b=\Big(\dfrac{\sin93.824^{\circ}}{\sin29^{\circ}}\Big)15\approx30.871$
