Question

Find all solutions in the interval [0,2π) for cos(x - π/6) = 1 + cos(x + π/6). Show work, please and state in solution set.

Answer 1

you should know the expansions for
cos(A+B) and for cos(A-B)

cos(x-π/6) = cosxcos π/6 + sinxsin π/6
= (√3/2)x + (1/2)sinx

cos(x+ π/6) = (√3/2)cosx - (1/2)sinx

so ....
cos(x - π/6) = 1 + cos(x + π/6)
(√3/2)x + (1/2)sinx = 1 + (√3/2)cosx - (1/2)sinx

sinx = 1
x = π/2 , (look at the sine curve, there is only one place where it has a value of 1)

check: x = π/2 or 90°
LS = cos 60° = 1/2
RS = 1 + cos 120°
= 1 + (-1/2) = 1/2
= LS

my answer is correct