The idea

The sine rule

Why a/sin A = b/sin B holds in any triangle, choosing the sine rule by spotting a matching side–angle pair, turning it upside down to find an angle, and the ambiguous case where two triangles fit.

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Geometry & measures · Trigonometry (sine & cosine rules)

The sine rule

Why a/sin A = b/sin B holds in any triangle, choosing the sine rule by spotting a matching side–angle pair, turning it upside down to find an angle, and the ambiguous case where two triangles fit.

Why it works

No right angle, no SOH-CAH-TOA

A triangle has angles

40°

and

75°

, and the side facing the

40°

angle is

9

cm. Find the side facing the

75°

angle. Every trig tool you own so far demands a right angle — and this triangle hasn't got one. That's the gap the sine rule fills: it works in any triangle. This one is our running example, and by the end it will take two lines.

Drop a perpendicular and it falls out

In any triangle

ABC

, drop a height

h

from

C

onto

AB

. That one line manufactures two right-angled triangles, and

h

can be read from each: on the left

h = b\sin A

, on the right

h = a\sin B

. Same height, so

b\sin A = a\sin B

— and dividing both sides by

\sin A \sin B

gives the rule. Repeating from another vertex extends it around the triangle:

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

Keep reading — free

The rest of the explanation, plus 2 worked examples you step through move by move.

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