Sine Rule

The Sine Rule pairs each side with the angle opposite it: $\frac{a}{\sin (A)}=\frac{b}{\sin (B)}=\frac{c}{\sin (C)}$.

To use it you need at least one complete pair, an angle together with the side opposite that angle, plus one more piece of information (a second angle, or a second side) to find the unknown.

Without a complete opposite pair, none of the fractions can be worked out, and you need the Cosine Rule instead.

In the Sine Rule the capital letters $A$, $B$ and $C$ are the angles of the triangle, and the lower-case letters $a$, $b$ and $c$ are the sides opposite those angles.

So side $a$ is opposite angle $A$, side $b$ is opposite angle $B$ and side $c$ is opposite angle $C$.

It is that opposite pairing that makes the three fractions equal; pairing a side with the angle next to it breaks the rule.

The sine rule pairs each side with the angle opposite it: $\frac{a}{\sin A}=\frac{b}{\sin B}$.

Here $a=5$ cm is opposite $A=30^{\circ }$, and $b$ is opposite $B=70^{\circ }$, so $b=\frac{a\sin B}{\sin A}=\frac{5\times \sin 70^{\circ }}{\sin 30^{\circ }}$.

Since $\sin 70^{\circ }=0.9397$ and $\sin 30^{\circ }=0.5$, $b=\frac{5\times 0.9397}{0.5}=\frac{4.6985}{0.5}=9.3969$.

So $b\approx 9.4\text{ cm}$ (1 d.p.). It makes sense that $b>a$, because the larger angle ($70^{\circ }$) sits opposite the longer side.

The sine rule pairs each side with the angle opposite it: $\frac{a}{\sin A}=\frac{b}{\sin B}$.

Here $a=6$ cm is opposite $A=45^{\circ }$, and $b$ is opposite $B=60^{\circ }$, so $b=\frac{a\sin B}{\sin A}=\frac{6\times \sin 60^{\circ }}{\sin 45^{\circ }}$.

Since $\sin 60^{\circ }=0.8660$ and $\sin 45^{\circ }=0.7071$, $b=\frac{6\times 0.8660}{0.7071}=\frac{5.1962}{0.7071}=7.3485$.

So $b\approx 7.3\text{ cm}$ (1 d.p.). It makes sense that $b>a$, because the larger angle ($60^{\circ }$) sits opposite the longer side.

The sine rule pairs each side with the angle opposite it: $\frac{a}{\sin A}=\frac{b}{\sin B}$.

Here $a=4$ cm is opposite $A=30^{\circ }$, and $b$ is opposite $B=100^{\circ }$, so $b=\frac{a\sin B}{\sin A}=\frac{4\times \sin 100^{\circ }}{\sin 30^{\circ }}$.

An obtuse angle still has a positive sine: $\sin 100^{\circ }=0.9848$. With $\sin 30^{\circ }=0.5$, $b=\frac{4\times 0.9848}{0.5}=\frac{3.9392}{0.5}=7.8785$.

So $b\approx 7.9\text{ cm}$ (1 d.p.). Side $b$ is the longest side, as it faces the largest angle.

Use the sine rule in the form that puts the unknown on top: $\frac{\sin A}{a}=\frac{\sin B}{b}$, so $\sin A=\frac{a\sin B}{b}$.

Substituting, $\sin A=\frac{5\times \sin 120^{\circ }}{10}$. Since $\sin 120^{\circ }=0.8660$, $\sin A=\frac{5\times 0.8660}{10}=\frac{4.3301}{10}=0.4330$.

Then $A={\sin }^{-1}(0.4330)=25.6589^{\circ }$, so $A\approx 25.7^{\circ }$ (1 d.p.).

The calculator's second solution, $180^{\circ }-25.7^{\circ }=154.3^{\circ }$, is impossible here: angle $B$ is already $120^{\circ }$, so the other two angles must total only $60^{\circ }$.