30Btw, just to satiate my curiosity.. Whats the reward sir?[3] My chocolate is still pending! [2]
Prove the expansion of sin(A+B) in as many ways as you can...
Simpler proofs fetch higher reward ;)
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9 Answers
30
36sir i think maybe this can work...
consider a right triangle ABC
right angled at C
we drop a perpendicular from C on AB at D
clearly
LACD = B
NOW,
IN TRI ADC
SIN A = CD/AC , COS B = CD/AC
COS A = AD/AC, SIN B = AD/AC
THUS
SINA.COSB + COSA.SINB = 1
ALSO, A+B =90
SO, SIN(A+B) = 1
SO WE CAN CONCLUDE the exp of
SIN(A+B)
21Proof by calculus:
checkout post #6,#7,#8 here
http://www.targetiit.com/iit-jee-forum/posts/function-18569.html
39How about a complex numbers proof?
ei(a + b) = cos(a + b) + isin(a + b).
But ei(a + b) = eia + ib = eiaeib
= [cosa + isina][cosb + isinb]
= cosa.cosb + icosa.sinb + isina.cosb - sina.cosb
But this equals cos(a + b) + isin(a + b).
Equating imaginary parts,
sin(a + b) = sina.cosb + cosa.sinb
Note : This popped out of my head from nowhere...now I wish I'd used my head as much back in 11th and 12th..
62That proof from pritish is the first correct one...
Rahul your proof is true only for A+B = pi/2
I am hoping for a much more general proof...