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The Sine Rule

The sine rule connects each side of a triangle to the sine of its opposite angle. Use it whenever you have a matching side–angle pair, plus one more piece of information.

30 min Video by Zeeshan Zamurred Trigonometric Ratios

Edexcel AS Level Maths: 9.2 The Sine RuleWatch the full walkthrough before the notes below.

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What you'll be able to do

State the sine rule
Find a missing side using the sine rule
Find a missing angle using the sine rule
Understand the ambiguous (two-solution) case

The rule

Each side divided by the sine of its opposite angle gives the same value across the whole triangle.

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

Use the side/sine form to find a side; flip it for an angle.

Tip — Use the sine rule when you have a complete side–opposite-angle pair. Otherwise reach for the cosine rule.

Finding a side

Keep the unknown side on top. You need its opposite angle and one other complete side–angle pair.

\frac{a}{sin 4 0 ^{\circ}} = \frac{10}{sin 7 5 ^{\circ}}

a = \frac{10 sin 4 0 ^{\circ}}{sin 7 5 ^{\circ}}

Answer

a \approx 6.66

Finding an angle

Flip the rule so the sines are on top. You need a side and its opposite angle, plus the side opposite the unknown angle.

\frac{sin A}{a} = \frac{sin B}{b}

Sines on top when the unknown is an angle.

\frac{sin A}{8} = \frac{sin 4 0 ^{\circ}}{6}

sin A = \frac{8 sin 4 0 ^{\circ}}{6} \approx 0.857

Answer

A \approx 59. 0^{\circ}

The ambiguous case

When finding an angle, $sin^{- 1}$ on a calculator gives only the acute answer — but its obtuse partner $18 0^{\circ} - A$ may also fit the triangle. Always check whether the second angle is possible (the angles must still sum to less than $18 0^{\circ}$ ).

Tip — If the question gives a diagram showing an obtuse angle, or the acute answer makes the angle sum impossible, use 180° − A.

Formula recap

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

Sine rule (side form).

\frac{sin A}{a} = \frac{sin B}{b}

Flipped form for angles.

A or 18 0^{\circ} - A

The ambiguous case.

Common mistakes to avoid

Using the sine rule with two sides and the included angle.

That is the cosine rule. The sine rule needs a side–opposite-angle pair.

Forgetting the second possible angle (180° − A) when finding an angle.

Always consider the obtuse partner and check if it fits.

Key takeaways

Sine rule: a/sin A = b/sin B = c/sin C.
Unknown side → sides on top; unknown angle → sines on top.
You need a complete side–opposite-angle pair to use it.
When finding an angle, check the ambiguous obtuse case 180° − A.

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