Cosine Rule Worked ExamplesIGCSE Maths 0580

The formula gives $a^2$, not $a$, so a common error is to forget the square root. The June 2025 examiner report found that "a very common incorrect answer came from a misconception in calculating" the cosine rule. Work out $126\cos 50^{\circ}$ as a single term and subtract it from $130$. Working out $(130 - 126)\cos 50^{\circ}$ instead is wrong. Note the angle used must be the one opposite the side being found — the labelling in the formula is not arbitrary.

A negative cosine means an obtuse angle — do not treat the minus sign as an error. Unlike the sine rule, the cosine rule has no ambiguous case: it distinguishes acute from obtuse automatically. Sense-check: $a$ is the longest side, so $A$ must be the largest angle.

The cosine rule creates the matching pair the sine rule needs — that is the standard two-step pattern. Use the unrounded $PR$ in part (b); using $15.3$ shifts the angle slightly. On a multi-step cosine rule question, the June 2025 examiner report listed "premature rounding of values." as a cause of lost marks. Check the angles sum sensibly: $68 + 65.3 = 133.3$, leaving $46.7^{\circ}$, and the smallest angle should face the shortest side $PQ = 12$. ✓

The angle must be the one between the two sides used. The formula is on the formula list, yet the June 2025 examiner report noted that candidates "misquoted formulas such as the volume of a cone, the area of a triangle or the cosine rule." A useful upper bound: the area can never exceed $\frac{1}{2}ab = 31.5$, which happens only when the angle is $90^{\circ}$ — so $24.1$ is plausible. ✓

Subtracting the bearings gives the angle at the harbour directly — no need to resolve into components. Sense-check against the triangle inequality: the distance must lie between $10 - 6 = 4$ and $10 + 6 = 16$. ✓

You must state why the negative root is rejected. The sine rule would also work here ($a$ and $A$ are a matching pair), and is quicker — but this version tests whether you can handle the cosine rule when the unknown appears twice. Check with the sine rule: $\sin C = \frac{5.97 \sin 110^{\circ}}{9}$ gives $C \approx 38.5^{\circ}$, and $110 + 38.5 + 31.5 = 180$. ✓

Type $\cos 115^{\circ}$ straight into the calculator instead of copying a value you remember. The sign comes out correctly and you keep full accuracy. There is also a quick check: when the included angle is obtuse, $QR^2$ must be more than $6.5^2 + 4.8^2 = 65.29$. In error-spotting questions, name the mistake exactly ("used $\cos 115^{\circ}$ as positive"). "He used the wrong formula" does not earn the mark.

Numbers that work out to exact values are a sign that the question is meant to be done without a calculator. You need $\cos 60^{\circ} = \tfrac{1}{2}$ and $\sin 60^{\circ} = \tfrac{\sqrt{3}}{2}$, and the graphs give the values for $120^{\circ}$. A common error in (a) is to subtract before multiplying: $(89 - 80) \times \tfrac{1}{2} = 4.5$. In (b), the minus sign in $-\tfrac{1}{2}$ is what tells you the angle is obtuse. Angles $B$ and $D$ add up to $180^{\circ}$, so $A$, $B$, $C$ and $D$ lie on a circle.

The cosine rule gives the angles at the centres, and these link the triangle to the circle measures. The overlap is not a sector. It is two segments back to back on the common chord $PQ$. A common error is to use $\angle PAB$ instead of $\angle PAQ$ as the sector angle, which gives only $2.19\text{ m}^2$. Keep the angles unrounded in your calculator from (a) and (b) through to the end of (c). The March 2026 examiner report reminded candidates to "use more than 3 figures in their intermediate calculations".

On a question like this, the March 2025 examiner report noted that candidates who gave only the $1$ d.p. value as their most accurate answer were not awarded the accuracy mark. In a show that, the value $57.605\ldots$ is what earns it. The June 2025 report found that "a very common incorrect answer came from a misconception in calculating" the cosine rule. Working out $(7633 - 7008)\cos 52^{\circ}$ instead of $7633 - 7008\cos 52^{\circ}$ gives $LM = 19.6$, far too short to be the side facing a $52^{\circ}$ angle between sides of $48$ and $73$.

On a quadrilateral question like this, the June 2025 examiner report noted that in some cases candidates added the diagonal on as part of the perimeter. Here the diagonal is the pipe $PR$, and adding it gives $63.6\text{ m}$. The same report found that some candidates lost accuracy by using a rounded length from an earlier part, and it listed "premature rounding of values." as a cause of lost marks. Using $PR = 16.3$ and $QR = 8.41$ gives a total of $47.2\text{ m}$, so keep full calculator values until the final answer.

This needs you to picture three dimensions from a plan. On a question like this, the March 2026 examiner report found that "Many made incorrect assumptions" about which distance to use. The horizontal distance comes from the cosine rule in the ground triangle, and the vertical drop is the difference in heights. A common error is to use the full $180\text{ m}$ instead of $145\text{ m}$, which gives $18.4^{\circ}$. Another is to use $QR$ or $QP$ as if it were the horizontal distance between the towers. Sketch the vertical right-angled triangle before you calculate.