56
PARALLÉLOGRAMME
résultante de trois puissances données elles-mêmes d’intensité et de direction.
On conclut encore de là
![{\displaystyle \left.{\begin{aligned}\operatorname {Sin} .x=&{\frac {1}{\Delta }}{\sqrt {B^{2}+C^{2}+2BC\operatorname {Cos} .a-(B\operatorname {Cos} .c-C\operatorname {Cos} .b)^{2}}},\\\\\operatorname {Sin} .y=&{\frac {1}{\Delta }}{\sqrt {C^{2}+A^{2}+2CA\operatorname {Cos} .b-(C\operatorname {Cos} .a-A\operatorname {Cos} .c)^{2}}},\\\\\operatorname {Sin} .z=&{\frac {1}{\Delta }}{\sqrt {A^{2}+B^{2}+2AB\operatorname {Cos} .c-(A\operatorname {Cos} .b-B\operatorname {Cos} .a)^{2}}}\,;\end{aligned}}\right\}\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba169f39cf00b04c403756a562437a60353f09d6)
(5)
et par suite
![{\displaystyle \left.{\begin{aligned}\operatorname {Tang} .x=&{\frac {\sqrt {B^{2}+C^{2}+2BC\operatorname {Cos} .a-(B\operatorname {Cos} .c-C\operatorname {Cos} .b)^{2}}}{A+B\operatorname {Cos} .c+C\operatorname {Cos} .b}},\\\\\operatorname {Tang} .y=&{\frac {\sqrt {C^{2}+A^{2}+2CA\operatorname {Cos} .b-(C\operatorname {Cos} .a-A\operatorname {Cos} .c)^{2}}}{B+C\operatorname {Cos} .a+A\operatorname {Cos} .c}},\\\\\operatorname {Tang} .z=&{\frac {\sqrt {A^{2}+B^{2}+2AB\operatorname {Cos} .c-(A\operatorname {Cos} .b-B\operatorname {Cos} .a)^{2}}}{C+A\operatorname {Cos} .b+B\operatorname {Cos} .a}}.\end{aligned}}\right\}\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/76267bad7cf7f6f89d7812cf7fc9f403833f1126)
(6)
Des équations (1) on tire
![{\displaystyle \left.{\begin{aligned}A=&\Delta .{\tfrac {\left(1-\operatorname {Cos} .^{2}a\right)\operatorname {Cos} .x-(\operatorname {Cos} .c-\operatorname {Cos} .a\operatorname {Cos} .b)\operatorname {Cos} .y-(\operatorname {Cos} .b-\operatorname {Cos} .c\operatorname {Cos} .a)\operatorname {Cos} .z}{1-\operatorname {Cos} .^{2}a-\operatorname {Cos} .^{2}b-\operatorname {Cos} .^{2}c+2\operatorname {Cos} .a\operatorname {Cos} .b\operatorname {Cos} .c}}\\B=&\Delta .{\tfrac {\left(1-\operatorname {Cos} .^{2}b\right)\operatorname {Cos} .y-(\operatorname {Cos} .a-\operatorname {Cos} .b\operatorname {Cos} .c)\operatorname {Cos} .z-(\operatorname {Cos} .c-\operatorname {Cos} .a\operatorname {Cos} .b)\operatorname {Cos} .x}{1-\operatorname {Cos} .^{2}a-\operatorname {Cos} .^{2}b-\operatorname {Cos} .^{2}c+2\operatorname {Cos} .a\operatorname {Cos} .b\operatorname {Cos} .c}}\\C=&\Delta .{\tfrac {\left(1-\operatorname {Cos} .^{2}c\right)\operatorname {Cos} .z-(\operatorname {Cos} .b-\operatorname {Cos} .c\operatorname {Cos} .a)\operatorname {Cos} .x-(\operatorname {Cos} .a-\operatorname {Cos} .b\operatorname {Cos} .c)\operatorname {Cos} .y}{1-\operatorname {Cos} .^{2}a-\operatorname {Cos} .^{2}b-\operatorname {Cos} .^{2}c+2\operatorname {Cos} .a\operatorname {Cos} .b\operatorname {Cos} .c}}\end{aligned}}\right\}\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a84eb4d1b175e709664c56368812555f4ad4fb8)
(7)
substituant ces valeurs dans l’équation (2), il viendra, en divisant
par
chassant le dénominateur et transposant