Page:Bovier-Lapierre - Traité élémentaire de trigonométrie rectiligne 1868.djvu/66

 Les corrections sont expliquées en page de discussion

Calcul de A.

$\operatorname {tang} {\frac {\mathrm {A} }{2}}={\sqrt {\frac {(p-b)(p-c)}{p(p-a)}}}$.

$\log \operatorname {tang} {\frac {\mathrm {A} }{2}}={\frac {1}{2}}\left[\log(p-b)+\log(p-c)-\log p-\log(p-a)\right]$

${\begin{aligned}\log(p-b)&=1{,}44736\\\log(p-c)&=1{,}55925\\-\log p\qquad \ &={\overline {2}}{,}07475\\-\log(p-a)&={\overline {2}}{,}70051\end{aligned}}$

---

${\begin{aligned}2\log \operatorname {tang} {\frac {\mathrm {A} }{2}}={\overline {1}}{,}78187&\\\log \operatorname {tang} {\frac {\mathrm {A} }{2}}={\overline {1}}{,}89094&\\37^{\circ }52^{\prime }\ldots {\underline {\ldots 73}}&\\47^{\prime \prime }\;\ldots \;21&\end{aligned}}$

${\overline {{\frac {\mathrm {A} }{2}}=37^{\circ }52^{\prime }47^{\prime \prime }}}\qquad 60$

$\qquad \qquad \qquad \qquad \quad {\overline {1260}}|{\underline {26}}$

$\ \ \ \ \ \mathrm {A} =75^{\circ }45^{\prime }34^{\prime \prime }\qquad \quad \ \ |47$

Calcul de B.

$\operatorname {tang} {\frac {\mathrm {B} }{2}}={\sqrt {\frac {(p-a)(p-c)}{p(p-b)}}}$.

$\log \operatorname {tang} {\frac {\mathrm {B} }{2}}={\frac {1}{2}}\left[{\color {Green}\log(p-a)+\log(p-c)}-\log p-\log(p-b)\right]$

Calcul de C.

$\operatorname {tang} {\frac {\mathrm {C} }{2}}={\sqrt {\frac {(p-a)(p-b)}{p(p-c)}}}$.

$\log \operatorname {tang} {\frac {\mathrm {C} }{2}}={\frac {1}{2}}\left[\log(p-a)+\log(p-b)-\log p-\log(p-c)\right]$

En faisant les calculs indiqués, comme on a fait pour l’angle A, on trouve

$\mathrm {B} =57^{\circ }55^{\prime }24^{\prime \prime },\quad \mathrm {C} =46^{\circ }19^{\prime }\ 0^{\prime \prime }$.

La somme des trois angles trouvés ne diffère de 180° que de $2^{\prime \prime }.$

---