![{\displaystyle \operatorname {Sin} .s={\frac {2p}{P}}\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/bb82cdb903a7cfddb1133f8ff53785cba4171de9)
(
xx)
![{\displaystyle \operatorname {Cos} .S=-{\frac {2P}{p}}\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/f10a8fc80d7bd5e13d51ac58c643435fdb899071)
(
XX)
![{\displaystyle \operatorname {Sin} .(s-a)={\frac {2p}{P}}\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/4deeccfe1c98b70ee8be30468738b552f91185d7)
(
xxi)
![{\displaystyle \operatorname {Cos} .(S-A)={\frac {2P}{p}}\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/47ad2e8ac71999e22e5f7b6babf50826572da4cd)
(
XXI)
Si l’on multiplie respectivement ces quatre valeurs par les quatre qui les précèdent immédiatement, on aura encore
![{\displaystyle \operatorname {Sin} .^{2}s=p\operatorname {Cot} .{\frac {1}{2}}A\operatorname {Cot} .{\frac {1}{2}}B\operatorname {Cot} .{\frac {1}{2}}C,\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc6395530404c4b7080d5db8fef8304c7e49e61)
(
xxii)
![{\displaystyle \operatorname {Cos} .^{2}S=P\operatorname {Tang} .{\frac {1}{2}}a\operatorname {Tang} .{\frac {1}{2}}b\operatorname {Tang} .{\frac {1}{2}}c,\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6518704fd0d56a629e3137eb289339680a37ca20)
(
XXII)
![{\displaystyle \operatorname {Sin} .^{2}(s-a)=p\operatorname {Cot} .{\frac {1}{2}}A\operatorname {Tang} .{\frac {1}{2}}B\operatorname {Tang} .{\frac {1}{2}}C,\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c96a5019fd570795a7df1c50af1b3e7e8255dcd3)
(
xxiii)
![{\displaystyle \operatorname {Cos} .^{2}(S-A)=P\operatorname {Tang} .{\frac {1}{2}}a\operatorname {Cot} .{\frac {1}{2}}b\operatorname {Cot} .{\frac {1}{2}}c.\qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f05c9ca113a3d4a54db41fbb8af6ebd8ddf9ef9d)
(
XXIII)
Les formules (ii, II) reviennent (7) à
![{\displaystyle {\begin{aligned}{\frac {\operatorname {Sin} .a\operatorname {Cos} .{\frac {1}{2}}B}{\operatorname {Sin} .{\frac {1}{2}}A}}={\frac {\operatorname {Sin} .b\operatorname {Cos} .{\frac {1}{2}}A}{\operatorname {Sin} .{\frac {1}{2}}b}},&{\frac {\operatorname {Sin} .a\operatorname {Cos} .{\frac {1}{2}}C}{\operatorname {Sin} .{\frac {1}{2}}A}}={\frac {\operatorname {Sin} .c\operatorname {Cos} .{\frac {1}{2}}A}{\operatorname {Sin} .{\frac {1}{2}}C}},\\\\{\frac {\operatorname {Sin} .A\operatorname {Cos} .{\frac {1}{2}}b}{\operatorname {Sin} .{\frac {1}{2}}a}}={\frac {\operatorname {Sin} .B\operatorname {Cos} .{\frac {1}{2}}a}{\operatorname {Sin} .{\frac {1}{2}}b}},&{\frac {\operatorname {Sin} .A\operatorname {Sin} .{\frac {1}{2}}c}{\operatorname {Cos} .{\frac {1}{2}}a}}={\frac {\operatorname {Sin} .C\operatorname {Sin} .{\frac {1}{2}}a}{\operatorname {Cos} .{\frac {1}{2}}c}}\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4361a5139064d7f68b487e0d714711f6a6c87d8)
au moyen desquelles les formules (xiii), (XIII), (xiv), (XIV) prennent cette nouvelle forme