Question

For the equation cos (2x)= 1, the sum of the solutions in the interval [0,2π] is and why?

A) π/8
B) 3π/4
C) 3π/2
D) 3π
E) 7π/2

Answer 1

To find the solutions of the equation cos(2x) = 1 in the interval [0, 2π], you need to first find where cos(2x) equals 1.

Cosine has a maximum value of 1, which occurs when the angle is 0 degrees or a multiple of 360 degrees (0, 360, 720, ...). In radians, this corresponds to 0, 2π, 4π, ...

So, the solutions for 2x are:

2x = 0
2x = 2π
2x = 4π

Now, solve for x:

x = 0/2 = 0
x = 2π/2 = π
x = 4π/2 = 2π

So, the solutions in the interval [0, 2π] are x = 0, x = π, and x = 2π.

Now, calculate the sum of these solutions:

0 + π + 2π = 3π

The sum of the solutions in the interval [0, 2π] is 3π, which is option (D).