Page:Bovier-Lapierre - Traité élémentaire de trigonométrie rectiligne 1868.djvu/62

gles, et l’on n’a à chercher que deux nouveaux logarithmes au lieu de trois.

Exemple. — Dans un triangle on a

a = 215^m,36,xxxb = 183^m,47,xxxC = 71° 25′ 14″ ;

calculer les angles A et B, et le côté c.

Calcul de A et B.

$\operatorname {tang} {\frac {\mathrm {A-B} }{2}}\ =\ \operatorname {tang} {\frac {\mathrm {A+B} }{2}}\times {\frac {a-b}{a+b}}$.

$\log \operatorname {tang} {\frac {\mathrm {A-B} }{2}}\ =\ \log \operatorname {tang} {\frac {\mathrm {A+B} }{2}}+\log(a-b)-\log(a+b)$.

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$\mathrm {A+B} =180^{\circ }-\mathrm {C} =108^{\circ }34^{\prime }46^{\prime \prime }$,xxx$a+b=398{,}83$,

${\frac {\mathrm {A+B} }{2}}=54^{\circ }17^{\prime }23^{\prime \prime }$.                          $a-b=31{,}89$.

${\begin{aligned}\log \operatorname {tang} {\frac {\mathrm {A+B} }{2}}&=0{,}14336\\\log(a-b)&=1{,}50365\\-\log(a+b)&={\overline {3}}{,}39921\end{aligned}}$

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${\begin{aligned}\log \operatorname {tang} {\frac {\mathrm {A-B} }{2}}={\overline {1}}{,}04622\\6^{\circ }20^{\prime }\ldots \ldots \ldots 528&\\{\frac {}{\qquad \quad }}&\\49^{\prime \prime }\ \ldots \ldots \ 94&\\60&\end{aligned}}$

${\overline {\mathrm {\frac {A-B}{2}} \ =\ 6^{\circ }20^{\prime }49^{\prime \prime }}}\qquad {\overline {5640}}|{\frac {115}{\ 49}}$

${\begin{aligned}\mathrm {A} =\mathrm {\frac {A+B}{2}} +\mathrm {\frac {A-B}{2}} &=60^{\circ }38^{\prime }12^{\prime \prime }\\\mathrm {B} =\mathrm {\frac {A+B}{2}} -\mathrm {\frac {A-B}{2}} &=47^{\circ }56^{\prime }34^{\prime \prime }.\\\end{aligned}}$