![{\displaystyle {\begin{aligned}&\operatorname {Cos} .s={\sqrt {1-\operatorname {Sin} .^{2}s}}={\frac {\sqrt {4\operatorname {Sin} .^{2}{\frac {1}{2}}A\operatorname {Sin} .^{2}{\frac {1}{2}}B\operatorname {Sin} .^{2}{\frac {1}{2}}C-P^{2}}}{2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}},\\\\&\operatorname {Sin} .S={\sqrt {1-\operatorname {Cos} .^{2}S}}={\frac {\sqrt {4\operatorname {Cos} .^{2}{\frac {1}{2}}a\operatorname {Cos} .^{2}{\frac {1}{2}}b\operatorname {Cos} .^{2}{\frac {1}{2}}c-p^{2}}}{2\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}}\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afc36d88de4477b129b45c8add7bb6647981a130)
à l’aide des formules (xxvii) et (XXVII), ces dernières deviendront
![{\displaystyle \operatorname {Cos} .s={\frac {1-\operatorname {Sin} .^{2}{\frac {1}{2}}A-\operatorname {Sin} .^{2}{\frac {1}{2}}B-\operatorname {Sin} .^{2}{\frac {1}{2}}C}{2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01657f825cb58d76f9f99f889110d2b16417aab3)
![{\displaystyle =-{\frac {1-\operatorname {Cos} .A-\operatorname {Cos} .B-\operatorname {Cos} .C}{4\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}},\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1fb8ebd96f0a7190e1dd5cb251a7b89d46f225)
(
xxix)
![{\displaystyle \operatorname {Sin} .S=-{\frac {1-\operatorname {Cos} .^{2}{\frac {1}{2}}a-\operatorname {Cos} .^{2}{\frac {1}{2}}b-\operatorname {Cos} .^{2}{\frac {1}{2}}c}{2\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc177e63f352dd625a139a2b9cbaeaacca52110)
![{\displaystyle ={\frac {1+\operatorname {Cos} .a+\operatorname {Cos} .b+\operatorname {Cos} .c}{4\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}}.\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/116789ea144dd28faa2f004d50d07b72dec4ab7b)
(
XXIX)
On a aussi, par les formules (xviii) et (XVIII),
![{\displaystyle {\begin{aligned}&\operatorname {Cos} .(s-a)={\sqrt {1-\operatorname {Sin} .^{2}(s-a)}}={\frac {\sqrt {4\operatorname {Sin} .^{2}{\frac {1}{2}}A\operatorname {Cos} .^{2}{\frac {1}{2}}B\operatorname {Cos} .^{2}{\frac {1}{2}}C-P^{2}}}{2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}},\\\\&\operatorname {Sin} .(S-A)={\sqrt {1-\operatorname {Cos} .^{2}(S-A)}}={\frac {\sqrt {4\operatorname {Cos} .^{2}{\frac {1}{2}}a\operatorname {Sin} .^{2}{\frac {1}{2}}b\operatorname {Sin} .^{2}{\frac {1}{2}}c-p^{2}}}{2\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}}\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6713d3ba2f196cc89891dbfe8b8dc75dc24ce12)
à l’aide des formules (xxviii) et (XXVIII), ces dernières deviendront
![{\displaystyle \operatorname {Cos} .(s-a)={\frac {1+\operatorname {Sin} .^{2}{\frac {1}{2}}A-\operatorname {Sin} .^{2}{\frac {1}{2}}B-\operatorname {Sin} .^{2}{\frac {1}{2}}C}{2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27b96318bc1356b52343ca8fb384f22f88c9a228)
![{\displaystyle ={\frac {1-\operatorname {Cos} .A+\operatorname {Cos} .B+\operatorname {Cos} .C}{4\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb26bc2ba205f9885029aed105218d81a1fffd3d)
(
xxx)
![{\displaystyle \operatorname {Sin} .(S-A)={\frac {1+\operatorname {Cos} .^{2}{\frac {1}{2}}a-\operatorname {Cos} .^{2}{\frac {1}{2}}b-\operatorname {Cos} .^{2}{\frac {1}{2}}c}{2\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffbe7544078ff25d64cb4b037c3241985ba9fda1)
![{\displaystyle ={\frac {1+\operatorname {Cos} .a-\operatorname {Cos} .b-\operatorname {Cos} .c}{4\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}}.\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/42eb9d1fe9dc2a5b7891244f5e6880ab2ba8c114)
(
XXX)
La comparaison des formules (xvii), (XVII), (xviii), (XXIII) aux formules (xxix), (XXIX), (xxx), (XXX), donne encore
![{\displaystyle \operatorname {Tang} .s=-{\frac {2P}{1-\operatorname {Cos} .A-\operatorname {Cos} .B-\operatorname {Cos} .C}},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a3ed8c6a2a74b44084e7b0c522db865e33df1a)
(
xxxi)