Question #23771

2 Answers
Shiva Prakash M V · Aviv S.
Feb 11, 2018

Equating the coefficients of real parts, we get the value of #cos100x#.
Equating the coefficients of imaginary parts, we get the value of #sin100x#.

Explanation:

Given:
#cosx=1/10#
#cos100x=?#

By DeMoivre's theorem,
#(cosx+isinx)^100=cos100x+isin100x#

A complex number has a real part and an imaginary part.
The term associated with #i# is termed as the imaginary part.
Expanding,
#(cosx+isinx)^100= 100C0cos^100x(isinx)^0+ 100C1cos^99x(isinx)^1+ 100C2cos^98x(isinx)^2+......... 100C98cos^2x(isinx)^98+ 100C99cos^1x(isinx)^99+ 100C100cos^0x(isinx)^100#

Noting that:
#i# is an imaginary number defined by #i=sqrt(-1)#
#i^0=1#
#i^1=i#
#i^2=-1#
#i^3=-i#
In the above expansion, we get several terms containing the imaginary number #i#.

Equating the coefficients of real parts, we get the value of #cos100x#.
Equating the coefficients of imaginary parts, we get the value of #sin100x#.

Nghi N
Feb 12, 2018

cos 100x = - 0.83

Explanation:

cos x = 1/10 = 0.10
Calculator gives --> #x = +- 84^@26#
#100x = +- 8,426^@#
#cos +- 8,426 = cos +- [146 + (23*360)] = #
#= cos +- (146 + 8280) = cos +- 146^@ #
Calculator gives -->
#cos 100x = cos +- 146 = - 0.83#

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