Sine Rule

To apply the Sine Rule, you need to know at least one pair of an angle and its opposite side in the triangle.

In the Sine Rule formula, $a$, $b$, and $c$ represent the sides of the triangle, and $A$, $B$, and $C$ are the angles opposite those sides.

Using the Sine Rule, $\frac{a}{\sin (A)}=\frac{b}{\sin (B)}$. Substituting the known values, $\frac{5}{\sin (30^{\circ })}=\frac{b}{\sin (70^{\circ })}$. Since $\sin (30^{\circ })=0.5$ and $\sin (70^{\circ })\approx 0.94$, we cross-multiply to get $b\approx \frac{5\times 0.94}{0.5}\approx 9.4\text{ cm}$.

Using the Sine Rule, $\frac{a}{\sin (A)}=\frac{b}{\sin (B)}$. Substituting the known values, $\frac{6}{\sin (45^{\circ })}=\frac{b}{\sin (60^{\circ })}$. Since $\sin (45^{\circ })\approx 0.7$ and $\sin (60^{\circ })\approx 0.9$, we cross-multiply to get $b\approx \frac{6\times 0.9}{0.7}\approx 7.7\text{ cm}$.

Using the Sine Rule, $\frac{a}{\sin (A)}=\frac{b}{\sin (B)}$. Substituting the known values, $\frac{4}{\sin (30^{\circ })}=\frac{b}{\sin (100^{\circ })}$. Since $\sin (30^{\circ })=0.5$ and $\sin (100^{\circ })\approx 1.0$, we cross-multiply to get $b\approx \frac{4\times 1.0}{0.5}\approx 8.0\text{ cm}$.

Using the Sine Rule, $\frac{a}{\sin (A)}=\frac{b}{\sin (B)}$. Substituting the known values, $\frac{5}{\sin (A)}=\frac{10}{\sin (120^{\circ })}$. Since $\sin (120^{\circ })\approx 0.866$, we get $\sin (A)\approx \frac{5\times 0.866}{10}\approx 0.433$. Finally, using the inverse sine function, we find that $A\approx 25.8^{\circ }$.