Determine the point(s) of intersection of the graphs of y=cosx and y=cos2x on the interval xE [0,2pi]

Answer 1

Determine the point(s) of intersection of the graphs of y = cos x and y = cos 2x on the interval xE [0, 2π]

Detailed Solution:

y = cos x and y = cos 2x [0, 2π]

At intersection point, they are equal:

cos 2x = cos x==> 2 cos² x – 1 = cos x==>2 cos² x – cos x – 1 = 0==>(cos x – 1)(2 cos x + 1) = 0==>

cos x – 1 = 0 ==> cos x = 1 ==> x = cos^-1 1 ==> x = 0

2 cos x + 1 = 0 ==> 2 cos x = – 1 ==> cos x = – 1/2 (Q2 & Q3)

Quadrant 2: cos x = –1/2 ==> x = cos^-1 (–1/2) ==> x = π – π/3 ==> x = 2π/3

Quadrant 3: cos x = –1/2 ==> x = cos^-1 (–1/2) ==> x = π + π/3 ==> x = 4π/3

Therefore, intersection point are at: x = {0, 2π/3, 4π/3}

Answer 2

cos x = cos² x

0 = cos² x - cos x

Factor and use Zero Product Rule.