The sine rule and the area formula

Use the sine rule to find sides and angles in any triangle, watch for the ambiguous case, and find area with ½ab sin C.

The sine rule

In triangle $ABC$, side $a$ is opposite angle $A$, side $b$ opposite $B$ and side $c$ opposite $C$.

Sine rule

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

Use it when you know a side and its opposite angle, plus one more side or angle.

Example 1: a side●

In $\triangle ABC$, $A = 40^\circ$, $B = 65^\circ$ and $a = 10$. Find $b$ to 1 decimal place.

$\dfrac{b}{\sin 65^\circ} = \dfrac{10}{\sin 40^\circ}$

$b = \dfrac{10\sin 65^\circ}{\sin 40^\circ} \approx 14.1$

Finding an angle: the ambiguous case

To find an angle, flip the rule: $\dfrac{\sin A}{a} = \dfrac{\sin B}{b}$. But $\sin\theta = k$ has an acute solution and an obtuse one ($180^\circ$ minus it). Both may give a valid triangle.

Example 2●●●

In $\triangle ABC$, $a = 8$, $b = 10$ and $A = 40^\circ$. Find $B$ to the nearest degree.

$\sin B = \dfrac{10\sin 40^\circ}{8} \approx 0.803$

$B \approx 53^\circ$ or $B \approx 180^\circ - 53^\circ = 127^\circ$

Both are possible: two different triangles fit the data.

Check the angle sum: 40° + 127° = 167° is less than 180°, so both work.

The area formula

Area of a triangle

$$\text{Area} = \tfrac12 ab\sin C$$

Use two sides and the angle between them (the included angle).

Two sides and the angle between them

Drag B along the axis and drag C. The area ½ab sin C is the same as ½ × base × height, because the height is AC sin A.

Try this

Keep $C$ the same distance from $A$ and swing it round. When is the area largest?

Example 3●

Find the area of a triangle with sides $7$ and $9$ and included angle $50^\circ$.

Area $= \tfrac12 \times 7 \times 9 \times \sin 50^\circ \approx 24.1$ square units

Flashcards

Sine rule

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

When to use the sine rule

A side and its opposite angle are both known.

Ambiguous case

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

Area

$\tfrac12 ab\sin C$ with $C$ the included angle.

Practice

$A = 30^\circ$, $a = 6$ and $B = 45^\circ$. Find $b$ to 1 decimal place.
Find the exact area of a triangle with sides $5$ and $8$ and included angle $120^\circ$. (Type something like 4sqrt(2).)
Which information suits the sine rule?
A triangle has area $20$ and two sides $6$ and $8$. Find the possible included angles to the nearest degree.
$\sin B = 0.9$ from the sine rule, and $A = 100^\circ$. Then $B$ is:
Find the area of a triangle with sides $10$ and $10$ and included angle $30^\circ$.
In the sine rule, side $a$ is:
$\tfrac12 ab\sin C$ is largest, for fixed $a$ and $b$, when $C =$