Page:Bovier-Lapierre - Traité élémentaire de trigonométrie rectiligne 1868.djvu/75

Calcul de $\varphi$.

$\log \operatorname {tang} \varphi =\log b-\log a$

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$\ \;\log b=1{,}80240$

$\ \;\log a=2{,}12359$

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$\log \operatorname {tang} \varphi ={\overline {1}}{,}67881\quad \ \$

$\qquad \ \ \ \varphi =25^{\circ }30^{\prime }58^{\prime \prime }$

Calcul de A et de B dans le triangle ABC.

$\log \operatorname {tang} {\frac {\mathrm {A-B} }{2}}=\log \operatorname {tang} {\frac {\mathrm {A+B} }{2}}+\log \operatorname {tang} (45^{\circ }-\varphi )$

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${\frac {\mathrm {A+B} }{2}}=90^{\circ }-{\frac {\mathrm {ACB} }{2}}=69^{\circ }16^{\prime }32^{\prime \prime }$

$45^{\circ }-\varphi =19^{\circ }29^{\prime }2^{\prime \prime }$

$\log \operatorname {tang} {\frac {\mathrm {A+B} }{2}}=0{,}42210$

$\log \operatorname {tang} (45^{\circ }-\varphi )={\overline {1}}{,}54877$

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$\log \operatorname {tang} {\frac {\mathrm {A-B} }{2}}={\overline {1}}{,}97087$

$\qquad \qquad {\frac {\mathrm {A-B} }{2}}=43^{\circ }4^{\prime }48^{\prime \prime }$

$\qquad \qquad \qquad \quad \ \ \mathrm {A} =112^{\circ }21^{\prime }20^{\prime \prime }$

$\qquad \qquad \qquad \ \ \ \ \;\mathrm {B} =\ \ 26^{\circ }11^{\prime }44^{\prime \prime }$

Calcul de c dans le triangle ABC.

$\log c=\log a+\log \sin \mathrm {C} -\log \sin \mathrm {A}$

${\begin{aligned}\log a&=2{,}12359\\\log \sin \mathrm {C} &={\overline {1}}{,}82083\\-\log \sin \mathrm {A} &=0{,}03393\end{aligned}}$

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${\begin{aligned}\qquad \ \ \ \log c&=\ 1{,}97835\\c&=95^{\mathrm {m} }{,}137\end{aligned}}$