Question

Rewrite cos(x+4pi/3) in terms of sin(x) and cos(x)

Answer 1

x8659-3358468854---6666x

Answer 2

`(√(3) )/(2)` sinx -
`(1)/(2)` cosx

Explanation:

using the sum identity for cosine

cos(a + b) = cosacosb - sinasinb

then

cos(x +
`(4\pi )/(3)` )

= cosxcos(
`(4\pi )/(3)`) - sinxsin(
`(4\pi )/(3)` )

[
`(4\pi )/(3)` is an angle in the third quadrant where sin and cosine < 0 ]

[ the related acute angle to
`(4\pi )/(3)` =
`(4\pi )/(3)` - π =
`(\pi )/(3)` ]

= cosx × - cos(
`(\pi )/(3)` ) - sinx × - sin(
`(\pi )/(3)` )

= cosx × -
`(1)/(2)` - sinx × -
`(√(3) )/(2)`

= -
`(1)/(2)` cosx +
`(√(3) )/(2)` sinx

=
`(√(3) )/(2)` sinx -
`(1)/(2)` cosx