Question

cho tam giác ABC , chứng minh rằng :

a) \(\sin\)(B + C) =  \(\sin\)A 

b) \(\cos\)(A + B) = -\(\cos\)C

c) \(\sin\)\(\frac{B+C}{2}\) =  \(\cos\)\(\frac{A}{2}\)

d) \(\tan\)\(\frac{A+C}{2}\) = \(\cot\) \(\frac{B}{2}\)

Answer 1

a) ta có : A+B+C=180=\(\pi\)

=>B+C= \(\pi\) - A

=> sin (B+C)=Sin(\(\pi\)-A)=SinA

b) tương tự:

cos( A+B)= Cos (\(\pi\)-C)=-cosC

c) ta có A+B+C =\(\pi\)=>\(\frac{A}{2}\)+\(\frac{B}{2}\)+\(\frac{C}{2}\)=\(\frac{\pi}{2}\)

=> sin (\(\frac{B+C}{2}\))=sin(\(\frac{\pi}{2}\)-\(\frac{A}{2}\))=cos(\(\frac{A}{2}\))

d) tương tự:

tan \(\frac{A+C}{2}\)=tan(\(\frac{\pi}{2}\)-\(\frac{B}{2}\))= cot\(\frac{B}{2}\)

===> đpcm