Question

At what x-values do the graphs of the functions y = cos 2x and y equals cosine squared x minus 1 intersect over the interval 0 less than or equals x less than 2 pi? Select all that apply.
(1 point)
Responses

negative Start Fraction pi over 2 End Fraction
Image with alt text: negative Start Fraction pi over 2 End Fraction

Start Fraction pi over 2 End Fraction
Image with alt text: Start Fraction pi over 2 End Fraction

π
π

Start Fraction 3 pi over 2 End Fraction

Answer 1

To find the x-values where the graphs of the functions \(y = \cos(2x)\) and \(y = \cos^2(x) - 1\) intersect over the interval \(0 \leq x < 2\pi\), we need to set the two equations equal to each other:

\[
\cos(2x) = \cos^2(x) - 1
\]

Using the identity \(\cos(2x) = 2\cos^2(x) - 1\), we can rewrite the equation:

\[
2\cos^2(x) - 1 = \cos^2(x) - 1
\]

Now, simplify this:

\[
2\cos^2(x) - 1 = \cos^2(x) - 1
\]
\[
2\cos^2(x) - \cos^2(x) = 0
\]
\[
\cos^2(x) = 0
\]

Taking the square root of both sides, we find:

\[
\cos(x) = 0
\]

The cosine function is equal to zero at:

\[
x = \frac{\pi}{2} + k\pi \quad \text{for } k \in \mathbb{Z}
\]

Within the interval \(0 \leq x < 2\pi\), the solutions are:

1. \(x = \frac{\pi}{2}\)
2. \(x = \frac{3\pi}{2}\)

So the x-values where the graphs \(y = \cos(2x)\) and \(y = \cos^2(x) - 1\) intersect in the interval \(0 \leq x < 2\pi\) are:

- \( \frac{\pi}{2} \)
- \( \frac{3\pi}{2} \)

Thus, the correct answers are:

- \( \frac{\pi}{2} \)
- \( \frac{3\pi}{2} \)