Find the zeros of the function in the interval. [-2π, 2π]

The zeroes of the f(x) = ± π/4 , ±3π/4 , ±5π/4 , ±7π/4 ⇒ answer (d)

Step-by-step explanation:

* The domain of the function f(x) = 1/2 cos(2x) is [-2π , 2π]

* That means -2π ≤ x ≤ 2π

∴ The domain of 2x is ⇒ -4π ≤ 2x ≤ 4π

* lets solve for 2x and then find the values of x

∵ f(x) = 1/2 cos(2x)

∴ The zeroes of the function means f(x) = 0

∴ 1/2 cos(2x)  = 0 ⇒ multiply both sides by 2

∴ cos(2x) = 0

* In the four quadrant the values of cosФ = 0 at the y-axis with angles

  π/2 , 3π/2 (anti-clockwise) , -π/2 , -3π/2 (clockwise) for 1 cycle,

  but we have 2 cycles for 2x, so we add 2π for every angle of

  2x, then we have other 4 angles 5π/2 , 7π/2 (anti-clockwise),

  -5π/2 , -7π/2 (clockwise)

∴ 2x = ± π/2 , ±3π/2 , ±5π/2 , ±7π/2

* To find the value of x divide all by 2

∴ x = ± π/4 , ±3π/4 , ±5π/4 , ±7π/4

* The zeroes of the f(x) = ± π/4 , ±3π/4 , ±5π/4 , ±7π/4

think the answer would beb