[tex]\begin{gathered} c\text{ = }\frac{3\text{ }\sqrt[]{2}}{2} \\ a\text{ = 10} \end{gathered}[/tex]Explanation:
For the smaller triangle:
angle = 45°
opposite = c
hypotenuse = 3
To get c, we will apply sine rule:
[tex]\begin{gathered} \sin \text{ 45}\degree\text{ = }\frac{opposite}{\text{hypotenuse}} \\ \sin \text{ 45}\degree\text{ = }\frac{c}{3} \\ c\text{ = 3(sin45}\degree)\text{ = 3(}\frac{\sqrt[]{2}}{2}) \\ c\text{ = }\frac{3\text{ }\sqrt[]{2}}{2} \end{gathered}[/tex]
For the larger triangle:
angle = 30°
opposite = 5
hypotenuse = a
To get a, we will apply sine rule:
[tex]\begin{gathered} \sin \text{ 30}\degree\text{ = }\frac{opposite}{hypotenuse} \\ \sin \text{ 30}\degree\text{ = }\frac{5}{a} \\ \text{cross multiply:} \\ a\text{ (sin 30}\degree)\text{ = 5} \\ a\text{ = }\frac{5}{\text{sin 30}\degree\text{ }} \\ a\text{ = }\frac{5}{\frac{1}{2}}\text{ = 5 }\times\frac{2}{1} \\ a\text{ = 10} \end{gathered}[/tex]
