Tangent Rule (Napier's Analogy)

10.4 The Tangent Rule (Napier's Analogy)

Theorem. In any $\triangle ABC$,

and cyclically for the other pairs of angles.

Proof. By the sine rule, $b=2R\sin B$ and $c=2R\sin C$, so

Using the sum-to-product formulas, $\sin B-\sin C=2\cos\frac{B+C}{2}\sin\frac{B-C}{2}$ and $\sin B+\sin C=2\sin\frac{B+C}{2}\cos\frac{B-C}{2}$. So

Since $A+B+C=180^\circ$, $\frac{B+C}{2}=90^\circ-\frac A2$, so $\cot\frac{B+C}{2}=\cot\left(90^\circ-\frac A2\right)=\tan\frac A2$. Hence

Why it is useful. Napier's analogy is the standard tool for the ASA/AAS "solve the triangle" problem when two angles and a side are known but the required quantity is a difference of angles or sides — historically this mattered because it let one work purely with tables of tangents and cotangents rather than looking up an awkward inverse sine or cosine.

Worked example (verification). For $a=13,b=14,c=15$: $\cos A=\frac{14^2+15^2-13^2}{2(14)(15)}=0.6\Rightarrow A\approx53.13^\circ$; similarly $B\approx59.49^\circ$, $C\approx67.38^\circ$. Then $\frac{B-C}{2}\approx-3.945^\circ$, so $\tan\frac{B-C}2\approx-0.069$. The right side gives $\frac{14-15}{14+15}\cot(26.565^\circ)=\left(-\frac1{29}\right)(2)\approx-0.069$ — the two sides agree.

This is the section Exercise 10(a) directly follows, since the exercise's rule-identity, area, and height-and-distance problems all draw on the sine/cosine/projection/tangent rules covered in §§10.1–10.4.

A second worked example (solving by ASA). Suppose $A=40^\circ$ and $B-C=20^\circ$ are known (an ASA-style variant). From $B+C=180^\circ-A=140^\circ$ and $B-C=20^\circ$, adding and subtracting the two equations gives $B=80^\circ,C=60^\circ$ directly — no tangent rule needed once both the sum and the difference are already known. The tangent rule's real value shows up in the reverse direction: if only $b,c,A$ are known (not $B-C$ itself), Napier's analogy computes $\tan\frac{B-C}2$ first, from which $B-C$ follows, and combined with $B+C=180^\circ-A$ (already known), both $B$ and $C$ are pinned down without ever needing the cosine rule. …