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Question Number 76790 by john santu last updated on 30/Dec/19

$$calculate\frac{sin{72}^o+sin{148}^o+sin{140}^o}{sin{36}^o\times sin{74}^o\times sin{70}^o}.$$

Commented by benjo 1/2 santuyy last updated on 31/Dec/19

$$sirmjsmaybeyourhelpmetosolvethisproblem$$

Answered by benjo 1/2 santuyy last updated on 30/Dec/19

$$considera+b+c={180}^o,$$$$sina=sin(b+c)$$$$sin2a=-sin(2b+2c)=-2sin(b+c)cos(b+c)$$$$leta={36}^o,b={74}^o,c={70}^o$$$$sin{36}^o=sin{74}^{o}cos{70}^o+cos{74}^{o}sin{70}^o$$$$$$

Answered by benjo 1/2 santuyy last updated on 30/Dec/19

$$4sir?$$

Answered by mind is power last updated on 27/Feb/20

$$sin\left(148\right)=sin\left(32\right)$$$$sin\left(140\right)=sin\left(40\right)$$$$sin\left(a\right)+sin\left(b\right)=2cos\left(\frac{a-b}{2}\right)sin\left(\frac{a+b}{2}\right)$$$$sin\left(32\right)+sin\left(40\right)=2sin\left(36\right)cos\left(4\right)$$$$sin\left(72\right)=2sin\left(36\right)cos\left(36\right)$$$$sin\left(32\right)+sin\left(40\right)+\sin \left(72\right)=2\sin \left(36\right)\cos \left(36\right)+2\sin \left(36\right)\cos \left(4\right)$$$$=2\sin \left(36\right)(\cos \left(36\right)+cos\left(4\right))=2sin\left(36\right)\left(2cos\left(20\right)cos\left(16\right)\right)$$$$=cos\left(70\right)=sin\left(20\right),cos\left(16\right)=sin\left(74\right)$$$$sin\left(72\right)+\sin \left(148\right)+sin\left(140\right)=4sin\left(36\right)sin\left(74\right)sin\left(70\right)$$$$\frac{sin\left(72\right)+sin\left(148\right)+\sin \left(140\right)}{\sin \left(36\right)\sin \left(74\right)\sin \left(70\right)}=4$$

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