SOLUTION:
First we calculate for a with cosines rule
[tex]\begin{gathered} a^2=b^2+c^2-2bcCosA \\ b=\text{ 29yd, c= 23yd, A= 99}\degree \\ a^2=(29)^2+(23)^2-2(29)(23)Cos(99) \\ a^2=841^{}+529^{}-1334\times(-0.1564) \\ a^2=841^{}+529^{}-1334\times(-0.1564) \\ a^2=841^{}+529^{}+208.68 \\ a^2=1578.686 \\ a=\text{ }\sqrt[]{1578.686} \\ a=\text{ 39.73} \end{gathered}[/tex]
Then we calculate angle B with sine rule
[tex]\begin{gathered} \frac{a}{\sin A}=\text{ }\frac{b}{\sin B} \\ A=\text{ 99}\degree,\text{ a= 39.73 yd, b= 29 yd} \\ \frac{39.73}{\sin 99}=\text{ }\frac{29}{\sin B} \\ \text{Cross multiplying} \\ \sin \text{ B= }\frac{29\sin 99}{39.73} \\ \sin \text{ B= }\frac{29\times0.987}{39.73} \\ \sin \text{ B= 0.720} \\ B=\text{ }\sin ^{-1}(0.720) \\ B=\text{ 46.05}\degree \end{gathered}[/tex]
Then we find the last angle C
[tex]\begin{gathered} A\text{ + B + C= 180}\degree \\ 99\text{ + 46.05}\degree\text{ + C= 180}\degree \\ C=\text{ 180}\degree-\text{ 99}\degree-\text{ 46.05}\degree \\ C=\text{ 34.95}\degree \end{gathered}[/tex]
Final answers:
Side
a= 39.7 yd (nearest tenth)
Angles
B= 46.1 degrees (nearest tenth)
C= 35.0 degrees (nearest tenth)
