Răspuns:
a) Din teorema sinusurilor
[tex]\dfrac{a}{\sin A} = \dfrac{b}{\sin B} =\dfrac{c}{\sin C} = k[/tex]
[tex]\Rightarrow a = k \cdot \sin A, b = k \cdot \sin B, c = k \cdot \sin C\\[/tex]
[tex]\boldsymbol {b \cdot \cos B + c \cdot \cos C} = k \cdot \sin B \cos B + k \cdot \sin C \cos C = \\[/tex]
[tex]= \dfrac{k}{2}\Big(\underbrace{2 \sin B \cos B}_{\sin 2B} + \underbrace{2 \sin C \cos C}_{\sin 2C}\Big) = \dfrac{k}{2}\Big(\sin 2B + \sin 2C\Big) \\[/tex]
[tex]^{A+B+C = \pi \Rightarrow B+C = \pi - A} \ ^{\Rightarrow} \ ^{ \sin (B+C) = \sin(\pi - A) = \sin A}\\[/tex]
[tex]= \dfrac{k}{2} \cdot 2\sin \dfrac{2B + 2C}{2} \cos \dfrac{2B - 2C}{2} = k \cdot\sin (B + C) \cos (B - C) \\[/tex]
[tex]= \underbrace{k \cdot \sin A}_{a} \cos(B-C) = \boldsymbol {a \cdot \cos(B - C)}\\[/tex]
b) Din teorema sinusurilor
[tex]\dfrac{\sin A}{a} = \dfrac{\sin B}{b} =\dfrac{\sin C}{c} = k[/tex]
[tex]\Rightarrow \sin A = ak, \sin B = bk, \sin C = ck[/tex]
Din formula diferenței:
[tex]\sin(B - C) = \sin B \cos C - \cos B \sin C = bk\cos C - bk \cos B = k(b\cos C - c \cos B)\\[/tex]
[tex]\sin(C - A) = \sin C \cos A - \cos C \sin A = ck\cos A - ak \cos C = k(c\cos A - a \cos C)\\[/tex]
[tex]\sin(A - B) = \sin A \cos B - \cos A \sin B = ak\cos B - bk \cos A = k(a\cos B - b \cos A)\\[/tex]
Am obținut suma:
[tex]\boldsymbol {a \cdot \sin(B - C) + b \cdot \sin(C - A) + c \cdot \sin(A - B)} = \\[/tex]
[tex]= ak\Big(b\cos C - c \cos B\Big) + bk\Big(c\cos A - a \cos C\Big) + ck\Big(a\cos B - b \cos A\Big) \\[/tex]
[tex]= k \Big(ab\cos C - ac\cos B + bc\cos A - ab\cos C + ac\cos B - bc\cos A\Big)[/tex]
[tex]= \boldsymbol {0}[/tex]
q.e.d.
✍ Formule utilizate:
[tex]\boxed{\boldsymbol{ 2 \cdot \sin \alpha \cdot \cos \alpha = \sin 2\alpha}}[/tex]
[tex]\boxed{\boldsymbol{ \sin \alpha + \sin \beta = 2 \sin \dfrac{\alpha + \beta}{2} \cos \dfrac{\alpha - \beta}{2} }}[/tex]
[tex]\boxed{\boldsymbol{ \sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta }}[/tex]
Teorema sinusurilor: În orice triunghi ABC avem:
[tex]\boxed{\boldsymbol{ \dfrac{a}{\sin A} = \dfrac{b}{\sin B} =\dfrac{c}{\sin C} = 2R}}[/tex]
Teorema sinusurilor stabilește relația dintre valorile laturilor unui triunghi și sinusurile unghiurilor dintre ele.
Alte relații trigonometrice https://brainly.ro/tema/11058857, https://brainly.ro/tema/10587511
