Question

On interval 0 ≤ x < 2π, where are the x-intercepts of y = cos(2x)?

StartFraction pi Over 2 EndFraction and StartFraction 3 pi Over 2 EndFraction
0, π, and 2π
StartFraction pi Over 2 EndFraction, pi and StartFraction 3 pi Over 2 EndFraction

Answer 1

The x-intercepts of y = cos(2x) on the interval 0 ≤ x < 2π are found at x = π/4, 3π/4, 5π/4, and 7π/4, which occur when cos(2x) = 0.

Step-by-step explanation:

The student is asking about the x-intercepts of the trigonometric function y = cos(2x) on the interval 0 ≤ x < 2π. The x-intercepts occur when the y-value is zero.

For the cosine function, this happens when the argument of the cosine, which is 2x in this case, equals π/2, 3π/2, etc., or any odd multiple of π/2.

However, due to the fact that we have cos(2x) instead of cos(x), the multiples will be different.

To find the values of x:

1. Set cos(2x) to 0: cos(2x) = 0.
2. Solve for 2x being odd multiples of π/2: 2x = π/2, 3π/2, 5π/2, ...
3. Divide by 2 to solve for x: x = π/4, 3π/4, 5π/4, ...

On the interval from 0 to 2π, these x-values are π/4, 3π/4, 5π/4, and 7π/4.

These are the x-intercepts of y = cos(2x).