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Question Number 188515 by mnjuly1970 last updated on 02/Mar/23

$$$$$$inA\stackrel{\mathrm{\Delta }}{BC}:a=3,b=6,c=7$$$$$$$$$$$$findthevalueof:$$$$$$$$E=(a+b)cos\left(C\right)+(b+c)cos\left(A\right)+(a+c)cos\left(B\right)=?$$$$$$

Answered by mr W last updated on 02/Mar/23

$$E=(a+b+c)(\cos A+\cos B+\cos C)-(a\cos A+b\cos B+c\cos C)$$$$=(a+b+c)(1+4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2})-R(\sin 2A+\sin 2B+\sin 2C)$$$$=(a+b+c)+4(a+b+c)\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}-4R\sin A\sin B\sin C$$$$=(a+b+c)+8R(\sin A+\sin B+\sin C)\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}-4R\sin A\sin B\sin C$$$$=(a+b+c)+8R\times 4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}-4R\sin A\sin B\sin C$$$$=(a+b+c)+4R\sin A\sin B\sin C-4R\sin A\sin B\sin C$$$$=a+b+c$$$$=3+6+7=16✓$$

Commented by manxsol last updated on 03/Mar/23

$$perpendicularprojection$$

Commented by manxsol last updated on 03/Mar/23

Commented by mnjuly1970 last updated on 03/Mar/23

$$thanksalotsirW$$

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