Sine and Cosine Rules

In right-angled triangles, basic trigonometry (SOH-CAH-TOA) works perfectly because one angle is always 90°. But the real world rarely cooperates: the triangle formed by two ships and a lighthouse, or the triangle a surveyor measures across hilly terrain, will almost never have a right angle. To handle these general triangles, mathematicians developed two powerful tools — the Sine Rule and the Cosine Rule — that connect any triangle's side lengths and angles without requiring a right angle. The Sine Rule states that in any triangle, each side divided by the sine of its opposite angle gives the same value. The Cosine Rule is a generalisation of Pythagoras' theorem: it reduces to a² = b² + c² when the included angle is 90°, and handles the other cases by adding or subtracting a correction term. Together, these two rules let you find every missing side and angle in any triangle, given enough information to define it uniquely. These rules are used constantly in navigation, architecture, engineering, and physics. Knowing which rule to apply — and when — is the key skill. Broadly: use the Sine Rule when you have a matching side-angle pair, and use the Cosine Rule when you have two sides and the included angle, or all three sides.

Key takeaways
  • Label the Triangle
  • The Sine Rule — When to Use It
  • The Ambiguous Case (SSA)
  • The Cosine Rule — When to Use It
  • After Finding the First Unknown

Label the Triangle

Always label a triangle consistently. Call the vertices A, B, C. The side opposite vertex A has length a, the side opposite B has length b, and the side opposite C has length c. The angles at vertices A, B, C are also called A, B, C respectively. This labelling is assumed by both rules.

The Sine Rule — When to Use It

Use the Sine Rule when you know: (i) two angles and any side (AAS or ASA), or (ii) two sides and a non-included angle (SSA — be careful, this can give two solutions). The rule says a/sin A = b/sin B = c/sin C. Rearrange to find the unknown. For example, to find side a: a = (b · sin A)/sin B.

The Ambiguous Case (SSA)

When using the Sine Rule with two sides and a non-included angle, sin⁻¹ can return two possible angles (one acute, one obtuse) that both satisfy the equation. You must check whether both give a valid triangle (angles summing to 180°). If the given angle is already obtuse, only one solution exists. Always check both possibilities and discard any that make the angle sum exceed 180°.

The Cosine Rule — When to Use It

Use the Cosine Rule when you know: (i) two sides and the included angle (SAS) — rearrange to find the opposite side; or (ii) all three sides (SSS) — rearrange to find any angle. The formula is a² = b² + c² − 2bc·cos A. To find angle A given all three sides, rearrange: cos A = (b² + c² − a²)/(2bc).

After Finding the First Unknown

Once you have found one unknown, the problem often becomes easier. If you now have a matching side-angle pair, switch to the Sine Rule for efficiency. Always find the third angle using the fact that A + B + C = 180°, and double-check your answer makes geometric sense (all angles positive, all sides positive).

The Key Equation

asin⁡A=bsin⁡B=csin⁡C(Sine Rule)\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \qquad \text{(Sine Rule)}
  • a, b, c — Side lengths (length (e.g. cm, m)): The three sides of the triangle, each labelled to match its opposite vertex.
  • A, B, C — Opposite angles (degrees or radians): The interior angle at each vertex, opposite to the side of the same letter.

The Key Equation

a2=b2+c2−2bccos⁡A(Cosine Rule)a^2 = b^2 + c^2 - 2bc\cos A \qquad \text{(Cosine Rule)}
  • a — Side opposite angle A (length): The side whose length you are calculating (or whose opposite angle you are finding).
  • b, c — The other two sides (length): The two sides that form angle A between them.
  • A — Included angle (degrees or radians): The angle between sides b and c; the angle directly opposite side a.

Worked Example

Problem. In triangle ABC, side b = 7 cm, side c = 10 cm, and angle A = 48°. Find side a and then angle B.

a≈7.44 cm,B≈44.4°a \approx 7.44 \text{ cm}, \quad B \approx 44.4°

Answer. Side a ≈ 7.44 cm, angle B ≈ 44.4°

Figure
Diagram
General Triangle with Sine and Cosine Rule LabelsABCcbaA=48°B≈44°C≈88°hSine Rule: a/sin A = b/sin B = c/sin CCosine Rule: a² = b² + c² − 2bc·cos Ab=7 cm, c=10 cm, A=48°, a≈7.44 cm
Origins · 1 minNasīr al-Dīn al-Tūsī and how this was discovered
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Nasīr al-Dīn al-Tūsī
1201–1274 · Persian

Al-Tūsī was the first mathematician to treat trigonometry as an independent discipline rather than a tool of astronomy. In his 1259 work 'Treatise on the Quadrilateral,' he systematically developed the Sine Rule for plane and spherical triangles, proved general forms that had previously been known only in special cases, and compiled the most complete trigonometric tables of his era. Al-Tūsī survived the Mongol sack of Baghdad in 1258 partly because he was respected as a mathematician and astronomer — he went on to direct the famous Maragheh Observatory built under Hulagu Khan's patronage.

  • Right Triangle Trigonometry (SOH-CAH-TOA) — The Sine and Cosine Rules extend right-triangle trigonometry to all triangles; understanding SOH-CAH-TOA is the essential prerequisite.
  • Area of a Triangle Using Trigonometry — The formula Area = ½ab·sin C uses the same side-angle labelling as the Sine and Cosine Rules and is often applied together with them.
  • The Unit Circle and Trigonometric Functions — Understanding why sin θ and cos θ are defined for any angle (including obtuse angles) is crucial for applying both rules correctly, especially in the ambiguous case.
  • The Pythagorean Theorem — The Cosine Rule is a direct generalisation of Pythagoras' theorem; setting A = 90° gives a² = b² + c², showing the connection explicitly.