§. III.
Recherche des sommes ou différences d’angles ou de côtés.
La combinaison des formules (41) avec les formules (v) et (vi) donne
![{\displaystyle {\frac {\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}{\operatorname {Sin} .{\frac {1}{2}}A}}={\frac {\operatorname {Sin} .s}{\operatorname {Sin} .a}},\qquad {\frac {\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}{\operatorname {Sin} .{\frac {1}{2}}A}}={\frac {\operatorname {Sin} .(s-a)}{\operatorname {Sin} .a}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/934fa058bc261cacf407a2fc1edb24eaa4be8b22)
c’est-à-dire,
![{\displaystyle \operatorname {Sin} .s={\frac {\operatorname {Sin} .a\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}{\operatorname {Sin} .{\frac {1}{2}}A}},\qquad \qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7701d18a39899e18f063f6f8d5f637aa9e7684d5)
(
xiii)
![{\displaystyle \operatorname {Sin} .(s-a)={\frac {\operatorname {Sin} .a\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}{\operatorname {Sin} .{\frac {1}{2}}A}}.\qquad \qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/982c8ff98abff9777e9539faa01a64eaf502b14b)
(
xiv)
La combinaison des formules (42) avec les formules (V) et (VI) donne ensuite
![{\displaystyle {\frac {\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}{\operatorname {Cos} .{\frac {1}{2}}a}}=-{\frac {\operatorname {Cos} .S}{\operatorname {Sin} .A}},\qquad {\frac {\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}{\operatorname {Cos} .{\frac {1}{2}}a}}={\frac {\operatorname {Cos} .(S-A)}{\operatorname {Sin} .A}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/141b45e87e5c2ceb83cd13d50c1db9305b1ef77a)
c’est-à-dire,
![{\displaystyle \operatorname {Cos} .s=-{\frac {\operatorname {Sin} .A\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}{\operatorname {Cos} .{\frac {1}{2}}a}},\qquad \qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e38ce0718fe3a93ba53e0d4fb2e03e5111f50cb)
(
XIII)
![{\displaystyle \operatorname {Cos} .(S-A)=+{\frac {\operatorname {Sin} .A\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}{\operatorname {Cos} .{\frac {1}{2}}a}}.\qquad \qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b68bc7b16d14b2f40c2fe27d73e44e0df6694a70)
(
XIV)
Mais les formules (viii, VIII) donnent (7)
![{\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 \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c832d94fc9fedf859a3c4c49dd59af60cdad2fa)
(
xv)