Question #23771

2 Answers
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.

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