Question

Question #23771

Answer 1

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`.

Answer 2

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`