sin ( A + B ) = 4/5 and sin ( A - B ) =5/13 prove that tan 2A = 63/16

Asked by arjun.srivastava15 | 30th Jun, 2014, 09:11: AM

Expert Answer:

G i v e n space t h a t space sin open parentheses A plus B close parentheses equals 4 over 5 space a n d space sin open parentheses A minus B close parentheses equals 5 over 13. N o w space c o n s i d e r space cos open parentheses A plus B close parentheses : cos open parentheses A plus B close parentheses equals sin open square brackets straight pi over 2 minus open parentheses A plus B close parentheses close square brackets S i n c e space 3 comma 4 comma 5 space i s space t h e space p y t h a g o e a n space t r i p l e t comma space w e space h a v e comma space cos open parentheses A plus B close parentheses equals 3 over 5... left parenthesis 1 right parenthesis S i m i l a r l y comma space sin open parentheses A minus B close parentheses : cos open parentheses A minus B close parentheses equals sin open square brackets straight pi over 2 minus open parentheses A plus B close parentheses close square brackets S i n c e space 5 comma space 12 comma space 13 space i s space t h e space p y t h a g o r e a n space t r i p l e t comma space w e space h a v e cos open parentheses A minus B close parentheses equals 12 over 13.... left parenthesis 2 right parenthesis W e space k n o w space t h a t comma space space sin open parentheses A plus B plus A minus B close parentheses equals sin open parentheses A plus B close parentheses cos open parentheses A minus B close parentheses plus sin open parentheses A minus B close parentheses cos open parentheses A plus B close parentheses rightwards double arrow sin open parentheses 2 A close parentheses equals sin open parentheses A plus B close parentheses cos open parentheses A minus B close parentheses plus sin open parentheses A minus B close parentheses cos open parentheses A plus B close parentheses rightwards double arrow sin open parentheses 2 A close parentheses equals 4 over 5 cross times 12 over 13 plus 5 over 13 cross times 3 over 5 space space space space space space space space space open square brackets f r o m space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space a n d space g i v e n space e q u a t i o n s close square brackets rightwards double arrow sin open parentheses 2 A close parentheses equals 48 over 65 plus 15 over 65 rightwards double arrow sin open parentheses 2 A close parentheses equals 63 over 65.... left parenthesis 3 right parenthesis rightwards double arrow cos open parentheses 2 A close parentheses equals square root of 1 minus open parentheses 63 over 65 close parentheses squared end root rightwards double arrow cos open parentheses 2 A close parentheses equals square root of fraction numerator 65 squared minus 63 squared over denominator 65 squared end fraction end root rightwards double arrow cos open parentheses 2 A close parentheses equals square root of 256 over 65 squared end root rightwards double arrow cos open parentheses 2 A close parentheses equals 16 over 65..... left parenthesis 4 right parenthesis W e space k n o w space t h a t comma space tan open parentheses 2 A close parentheses equals fraction numerator sin open parentheses 2 A close parentheses over denominator cos open parentheses 2 A close parentheses end fraction rightwards double arrow tan open parentheses 2 A close parentheses equals fraction numerator 63 over 65 over denominator 16 over 65 end fraction rightwards double arrow tan open parentheses 2 A close parentheses equals 63 over 16

Answered by Vimala Ramamurthy | 30th Jun, 2014, 10:12: AM