The sine rule

The sine rule for any triangle, finding unknown sides and angles, and the ambiguous case.

The rule

Sine rule

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Side $a$ is opposite angle $A$, and so on. Use it when you know a side and its opposite angle.

Example 1●●

In $\triangle ABC$, $A = 40^\circ$, $B = 65^\circ$ and $b = 12$. Find $a$.

$\dfrac{a}{\sin 40^\circ} = \dfrac{12}{\sin 65^\circ}$

$a = \dfrac{12\sin 40^\circ}{\sin 65^\circ} \approx 8.51$

Example 2●●●

$a = 8$, $A = 50^\circ$, $b = 10$. Find $B$.

$\sin B = \dfrac{10\sin 50^\circ}{8} \approx 0.9576$

$B \approx 73.2^\circ$ or $B \approx 180 - 73.2 = 106.8^\circ$. Both fit (angle sum under $180^\circ$), so there are two possible triangles: the ambiguous case.

Flashcards

Sine rule (sides)

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$

Sine rule (angles)

$\dfrac{\sin A}{a} = \dfrac{\sin B}{b}$

Ambiguous case

Finding an angle: check $180^\circ - \theta$ too.

Practice

$A = 40^\circ$, $B = 65^\circ$, $b = 12$. Find $a$ (2 d.p.).
$a = 8$, $A = 50^\circ$, $b = 10$. Find the acute value of $B$ (1 d.p.).
Same triangle: the obtuse value of $B$ (1 d.p.)?
$A = 70^\circ$, $C = 45^\circ$, $c = 15$. Find $a$ (2 d.p.).
$A = 30^\circ$, $a = 5$, $B = 90^\circ$. Find $b$.
Which information suits the sine rule?
$A = 40^\circ$, $B = 75^\circ$. Find $C$.
$a = 6$, $A = 90^\circ$, $b = 3$. Find $B$.