![{\displaystyle \operatorname {Cot} .S=-{\frac {2p}{1+\operatorname {Cos} .a+\operatorname {Cos} .b+\operatorname {Cos} .c}},\qquad \quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd0be25299818f596de4b39ce5274951879c288)
(
XXXI)
![{\displaystyle \operatorname {Tang} .(s-a)={\frac {2P}{1-\operatorname {Cos} .A+\operatorname {Cos} .B+\operatorname {Cos} .C}},\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/17a0e00ada8261c9b5aa406317e79313c87148fc)
(
xxxii)
![{\displaystyle \operatorname {Cot} .(S-A)={\frac {2p}{1+\operatorname {Cos} .a-\operatorname {Cos} .b-\operatorname {Cos} .c}}.\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eb8e6eb175269978dafb1ba2627b00bdf45cc5)
(
XXXII)
En formant les expressions analogues, relatives à ![{\displaystyle s-b,\ s-c,\ S-B,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23edb8cd48c50b8f54efead1e2e4579d915e73c0)
on en conclura
![{\displaystyle \operatorname {Cot} .(s-a)+\operatorname {Cot} .(s-b)+\operatorname {Cot} .(s-c)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e136f5396566cd5337f1563c401caad9097a7f9)
![{\displaystyle {\frac {3+\operatorname {Cos} .A+\operatorname {Cos} .B+\operatorname {Cos} .C}{2P}}={\frac {\operatorname {Cos} .^{2}{\frac {1}{2}}A+\operatorname {Cos} .^{2}{\frac {1}{2}}B+\operatorname {Cos} .^{2}{\frac {1}{2}}C}{P}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c869d4d63955adc55fa673850adbbdbd654d41d)
(
xxxiii)
![{\displaystyle \operatorname {Tang} .(S-A)+\operatorname {Tang} .(S-B)+\operatorname {Tang} .(S-C)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e06a838f2d0502b79c09eb9969fca86634deec6)
![{\displaystyle {\frac {3-\operatorname {Cos} .a-\operatorname {Cos} .b-\operatorname {Cos} .c}{2p}}={\frac {\operatorname {Sin} .^{2}{\frac {1}{2}}a+\operatorname {Sin} .^{2}{\frac {1}{2}}b+\operatorname {Sin} .^{2}{\frac {1}{2}}c}{p}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/367d33650933b0cbcc8927099f5a586277ea359e)
(
XXXIII)
Par les formules (5) et par la formule (xxix), on a
![{\displaystyle {\begin{aligned}&\operatorname {Sin} .{\frac {1}{2}}s={\sqrt {\frac {1-\operatorname {Cos} .s}{2}}}=\\&{\sqrt {-{\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}{4\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}}}},\\\\&\operatorname {Cos} .{\frac {1}{2}}s={\sqrt {\frac {1+\operatorname {Cos} .s}{2}}}=\\&{\sqrt {\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}{4\operatorname {Sin} .{\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/f68cbe96bec0cd53a4a06ef2f023f75c61d99357)
comparant ces formules aux formules (34) et (35), on pourra leur donner cette forme
![{\displaystyle \operatorname {Sin} .{\frac {1}{2}}s=\qquad \qquad \qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd2b4c18ca670f556a7098cbb85b6706e965766d)
(
xxxiv)
![{\displaystyle {\sqrt {-{\frac {\operatorname {Sin} .{\frac {1}{2}}\left({\frac {1}{2}}\varpi -S\right)\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-A)\right\}\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-B)\right\}\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-C)\right\}}{\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/1641c6d02ad4c76a72eea3772287598f82f271e3)
![{\displaystyle \operatorname {Cos} .{\frac {1}{2}}s=\qquad \qquad \qquad \qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6884ded90c787800ff25f1b5844408b985dd31d)
(
xxxv)
![{\displaystyle {\sqrt {\frac {\operatorname {Cos} .{\frac {1}{2}}\left({\frac {1}{2}}\varpi -S\right)\operatorname {Sin} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-A)\right\}\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-B)\right\}\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-C)\right\}}{\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/ac167e20fb0db4ce85845fc1e65351935f031c0b)
d’où encore