![{\displaystyle \operatorname {Sin} .A\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c={\frac {p}{2\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}},\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8acc42aa5594d579e5272e95c2074b6d62ed608e)
(
XV)
![{\displaystyle \operatorname {Sin} .a\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C={\frac {P}{2\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}},\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe068a370e617bd058ee1d226d21e5147fdc27e)
(
xvi)
![{\displaystyle \operatorname {Sin} .A\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c={\frac {p}{2\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}}.\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b92fc3548586c710069524fc8fc1799e3c26192)
(
XVI)
Introduisant ces valeurs dans les numérateurs des quatre dernières formules, elles deviendront
![{\displaystyle \operatorname {Sin} .s={\frac {P}{2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}},\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a927dbf8c21eccefc42abf32926386465abab608)
(
xvii)
![{\displaystyle \operatorname {Cos} .S=-{\frac {p}{2\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}},\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5cbe4ee87c769f8721f8204c3c27bad7952adc)
(
XVII)
![{\displaystyle \operatorname {Sin} .(s-a)={\frac {P}{2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}},\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/463347a4af26aeef9970d80cd9a2ccacc429a8cd)
(
xviii)
![{\displaystyle \operatorname {Cos} .(S-A)={\frac {p}{2\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}}.\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e897f2df9f23181ed742ca0ccbc290542efa852b)
(
XVIII)
Les formules (viii) et (VIII), divisées l’une par l’autre, donnent (ii)
![{\displaystyle {\frac {\operatorname {Sin} .A}{\operatorname {Sin} .a}}={\frac {p}{P}}.{\frac {\operatorname {Sin} .B}{\operatorname {Sin} .b}}.{\frac {\operatorname {Sin} .C}{\operatorname {Sin} .c}}={\frac {p}{P}}.{\frac {\operatorname {Sin} .^{2}A}{\operatorname {Sin} .^{2}a}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afde2c0b024f6a9a6c796d80c37e8122bba9782b)
c’est-à-dire,
![{\displaystyle {\frac {P}{p}}={\frac {\operatorname {Sin} .A}{\operatorname {Sin} .a}}.\qquad \qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/084fd8eda3a4ca8ed15d043cd1572ec1bb61fcdd)
(
xix, XIX)
en introduisant tour-à-tour les valeurs de
et
tirées de cette dernière formule, dans les formules (xiii), (XIII), (xiv), (XIV), et ayant égard à la formule (7), on trouvera