Rewrite the cosine function f(x)=cos(2x+π) as a sine function. Then write 1–2 sentences to explain each step you took to arrive at your answer.

Answer 1

To rewrite the cosine function \( f(x) = \cos(2x + \pi) \) as a sine function, we can use the identity \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \).

1. Substitute: We start with \( f(x) = \cos(2x + \pi) \).
2. Apply the cosine angle addition formula: We know that \( \cos(\theta + \pi) = -\cos(\theta) \), which implies that \( \cos(2x + \pi) = -\cos(2x) \).
3. Rewrite in terms of sine: Next, we can apply the identity for cosine in terms of sine: \( \cos(2x) = \sin\left(\frac{\pi}{2} - 2x\right) \). Therefore, \[ f(x) = -\sin\left(\frac{\pi}{2} - 2x\right). \]

So, the final form is: \[ f(x) = -\sin\left(\frac{\pi}{2} - 2x\right). \]

Explanation:

In the first step, I recognized that the cosine function could be transformed using its relationship with sine. I then applied the property of cosine and used standard angle identities to express \( \cos(2x + \pi) \) as a negative sine function, illustrating the transformation from cosine to sine while making sure to maintain the same functional input.