A triangular plot has two sides of 25 m and 30 m. The area is 300 m^2. (a)(i) Find the value of 90^. (b)(i) Find the perimeter for the smaller-angle case. (b)(ii) Find the perimeter for the larger-angle case.

(a)(i) Two angles. 12(25)(30)=300 => =(600)/(750)=0.8. =0.8. _1=53.1^. (a)(ii) _2=126.9^. (b)(i) Third side for each angle. For 53.1^: c^2=625+900-150053.1^=625 => c=25.0 m; perimeter =80.0 m. For 126.9^: c^2=625+900-1500126.9^ is approximately 2425 => c is approximately 49.24 m; perimeter is approximately 104.2 m. perimeter =80.0 m. (b)(ii) For 126.9^: c^2=625+900-1500126.9^ is approximately 2425 => c is approximately 49.24 m; perimeter is approximately 104.2 m. GDC: In degree mode evaluate sin^-1(0.8), then each cosine rule expression.

Cosine Rule (AI HL)

Q: What is the cosine rule formula?

c^2 = a^2 + b^2 - 2ab cos(C), where C is the angle opposite side c. It is given in the formula booklet.

The cosine rule links all three sides of a triangle to one of its angles, so it's the tool to reach for whenever the sine rule's angle-side pairing isn't available - two sides with the angle between them, or all three sides on their own. This page covers the formula, two worked examples, and the mistake that costs the most marks. It's part of the broader Geometry & Trigonometry topic.

11 questions on this sub-topic.

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The formula

Covered under IB syllabus reference SL3.2, alongside the sine rule and the \(\tfrac12ab\sin C\) area formula (this section excludes the sine rule's ambiguous case). Both rules sit in the formula booklet, so the job is spotting which one a question actually needs.

Cosine rule

\[c^2=a^2+b^2-2ab\cos C\]

Use with two sides and the included angle to find the third side, or with all three sides to find an angle.

✓ In the formula booklet

Sine rule

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

Needs an angle paired with its opposite side - if that pairing isn't there, it's a cosine rule question instead.

✓ In the formula booklet

Need the full syllabus wording, formula-booklet table and GDC steps? See Geometry & Trigonometry.

Worked examples

Easy

GDC

[3 marks]

Find the area of triangle \(ABC\) with \(a = 12\) cm, \(c = 8\) cm, \(B = 35^{\circ}\).

Worked solution

Area \(= \tfrac12 ac\sin B\) M1
\(= \tfrac12(12)(8)\sin 35^{\circ}\) A1
\(\approx 27.5\) units². A1

M1 \(\tfrac12 ac\sin B\) A1 Correct substitution A1 Correct answer of \(\approx27.5\)

Hard

GDC

[7 marks]

A triangular plot has two sides of 25 m and 30 m. The area is \(300\) m\(^2\).

(a)(i) Find the value of \(\theta<90^{\circ}\).
(a)(ii) Find the value of \(\theta>90^{\circ}\).
(b)(i) Find the perimeter for the smaller-angle case.
(b)(ii) Find the perimeter for the larger-angle case.

Worked solution

(a)(i) Two angles. \(\tfrac12(25)(30)\sin\theta=300\Rightarrow\sin\theta=\dfrac{600}{750}=0.8.\) M1
\(\sin\theta=0.8.\) A1
\(\theta_1=53.1^\circ.\) A1

(a)(ii) \(\theta_2=126.9^\circ.\) A1

(b)(i) Third side for each angle. For \(53.1^\circ\): \(c^2=625+900-1500\cos53.1^\circ=625\Rightarrow c=25.0\) m; perimeter \(=80.0\) m. For \(126.9^\circ\): \(c^2=625+900-1500\cos126.9^\circ\approx2425\Rightarrow c\approx49.24\) m; perimeter \(\approx104.2\) m. M1
perimeter \(=80.0\) m. A1

(b)(ii) For \(126.9^\circ\): \(c^2=625+900-1500\cos126.9^\circ\approx2425\Rightarrow c\approx49.24\) m; perimeter \(\approx104.2\) m. A1

GDC: In degree mode evaluate \(\sin^{-1}(0.8)\), then each cosine rule expression.

M1 Area equation A1 \(\sin\theta\) A1 \(\theta=53.1^\circ\) A1 \(\theta=126.9^\circ\) M1 Cosine rule A1 Perimeter \(80.0\) m A1 Perimeter \(104.2\) m

Common mistakes

Reaching for the sine rule when the cosine rule is needed. If you only know three sides, or two sides and the angle between them, there's no opposite angle-side pair to work with - that's the cosine rule's job.
Losing the obtuse solution. When \(\cos C\) comes out negative, \(C\) is obtuse - don't round it away or "correct" it back to an acute angle; the cosine rule (unlike the sine rule) gives a single, unambiguous angle.
Mislabelling which side is squared alone. In \(c^2=a^2+b^2-2ab\cos C\), \(C\) must be the angle between the two sides \(a\) and \(b\) that are multiplied together - swap the labelling and the substitution comes out wrong even with correct arithmetic.

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Quick answers

When should I use the cosine rule instead of the sine rule?

Use the cosine rule when you know two sides and the angle between them (to find the third side), or all three sides (to find an angle). The sine rule needs an angle paired with its opposite side, which the cosine rule doesn't require.

What is the cosine rule formula?

\(c^2 = a^2 + b^2 - 2ab\cos C\), where \(C\) is the angle opposite side \(c\). It's given in the formula booklet.

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