cos A. cos 2A. cos 3A = 1/4

Asked by abhishek_nayak98 | 15th May, 2014, 05:49: PM

Expert Answer:

G i v e n space t h a t comma space cos A cross times cos 2 A cross times cos 3 A equals 1 fourth W e space k n o w space t h a t comma 2 cos A cos B equals cos open parentheses A minus B close parentheses plus cos open parentheses A plus B close parentheses R e w r i t e space t h e space g i v e n space e q u a t i o n space a s comma 2 cos A cross times cos 3 A cross times cos 2 A equals 1 half rightwards double arrow open square brackets cos open parentheses 2 A close parentheses plus cos open parentheses 4 A close parentheses close square brackets cos 2 A equals 1 half... left parenthesis 1 right parenthesis A l s o comma space w e space k n o w space t h a t comma space cos 4 A equals 2 cos squared 2 A minus 1 T h e r e f o r e comma open square brackets cos open parentheses 2 A close parentheses plus cos open parentheses 4 A close parentheses close square brackets cos 2 A equals open square brackets 2 cos squared 2 A minus 1 plus cos 2 A close square brackets cos 2 A equals open square brackets 2 cos squared 2 A plus cos 2 A minus 1 close square brackets cos 2 A equals open square brackets 2 k squared plus k minus 1 close square brackets k comma space w h e r e space k equals cos 2 A equals 2 k cubed plus k squared minus k F r o m space e q u a t i o n space left parenthesis 1 right parenthesis comma open square brackets cos open parentheses 2 A close parentheses plus cos open parentheses 4 A close parentheses close square brackets cos 2 A equals 1 half equals 2 k cubed plus k squared minus k rightwards double arrow 4 k cubed plus 2 k squared minus 2 k minus 1 equals 0 rightwards double arrow open parentheses 2 k plus 1 close parentheses open parentheses 2 k squared minus 1 close parentheses equals 0 rightwards double arrow k equals minus 1 half comma space k squared equals 1 half rightwards double arrow 2 cos squared A minus 1 equals fraction numerator minus 1 over denominator 2 end fraction comma space 2 cos squared 2 A minus 1 equals 1 half rightwards double arrow cos squared A equals 1 fourth comma space cos squared 2 A equals 3 over 4 rightwards double arrow cos A equals plus-or-minus 1 half comma cos 2 A equals plus-or-minus fraction numerator square root of 3 over denominator 2 end fraction rightwards double arrow A equals cos to the power of minus 1 end exponent open parentheses 1 half close parentheses space o r space 2 A equals cos to the power of minus 1 end exponent open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses

Answered by Vimala Ramamurthy | 16th May, 2014, 10:03: AM