Using the sine rule and the cosine rule to find sides and angles in non-right-angled triangles, and using the formula one half ab sin C to find the area of a triangle.

What this dot point is asking

The SQA wants you to solve non-right-angled triangles. You should be able to choose and use the sine rule or the cosine rule to find a missing side or angle, and use the area formula $\tfrac{1}{2}ab\sin C$ to find the area of a triangle from two sides and the included angle.

Labelling a triangle

By convention, the angles are named with capital letters $A$, $B$, $C$ and the side opposite each angle takes the matching lower-case letter, so side $a$ is opposite angle $A$. Getting this pairing right is the key to using both rules.

The sine rule

The sine rule relates each side to the sine of its opposite angle. Use it when you have a complete side-angle pair (a side and its opposite angle) and one more measurement.

The cosine rule

The cosine rule handles the cases the sine rule cannot: two sides and the included angle, or all three sides.

The area of a triangle

When you know two sides and the angle between them, you can find the area directly, without the perpendicular height.

Choosing the right tool

A quick decision guide keeps you from reaching for the wrong rule. If the triangle is right-angled, use basic trigonometry (sine, cosine, tangent) or Pythagoras. If it is not right-angled, look at what you are given: a side opposite a known angle points to the sine rule; two sides with the angle between them, or all three sides, points to the cosine rule; two sides and the included angle for an area points to $\tfrac{1}{2}ab\sin C$.

Examples in context

These rules measure triangles that cannot be reached or are not right-angled, the staple of surveying and navigation. A surveyor finds the distance across a river by measuring a baseline and two angles, then applying the sine rule. A ship's navigator uses the cosine rule to find the direct distance between two legs of a journey when the turning angle is known. The area formula gives the size of a triangular plot from two boundary lengths and the angle between them.

Try this

Q1. Find side $c$ when $a = 5$, $b = 6$ and $C = 60^\circ$. [3 marks]

• Cue. $c^2 = 25 + 36 - 60\cos 60^\circ = 31$, so $c = 5.6$.

Q2. Find the area of a triangle with sides $8$ and $10$ and included angle $30^\circ$. [2 marks]

• Cue. $\tfrac{1}{2} \times 8 \times 10 \times \sin 30^\circ = 20$ cm$^2$.

Q3. Find angle $B$ when $b = 7$, $a = 9$ and $A = 50^\circ$. [3 marks]

• Cue. $\dfrac{\sin B}{7} = \dfrac{\sin 50^\circ}{9}$, so $B = 36.6^\circ$.