Question

Which of the following sine functions is equivalent to f(x)=cos(x)
?

Option #1: f(x)=sin(x−π2)


Option #2: f(x)=sin(x+π2)


Option #3: f(x)=sin(x+π)


(1 point)
Option #

Answer 1

To determine which sine function is equivalent to \( f(x) = \cos(x) \), we can use the phase shift property of the sine and cosine functions.

The cosine function can be expressed in terms of the sine function as follows:
\[
\cos(x) = \sin\left(x + \frac{\pi}{2}\right)
\]

Now, let's evaluate the options:

1. **Option #1: \( f(x) = \sin(x - \frac{\pi}{2}) \)**

Using the sine function's phase shift identity, we have:
\[
\sin(x - \frac{\pi}{2}) = -\cos(x)
\]
This is not equivalent to \( \cos(x) \).

2. **Option #2: \( f(x) = \sin(x + \frac{\pi}{2}) \)**

Using the sine function's phase shift identity:
\[
\sin(x + \frac{\pi}{2}) = \cos(x)
\]
This is equivalent to \( \cos(x) \).

3. **Option #3: \( f(x) = \sin(x + \pi) \)**

Using the sine function’s property:
\[
\sin(x + \pi) = -\sin(x)
\]
This is not equivalent to \( \cos(x) \).

Based on the evaluations, the correct answer is **Option #2: \( f(x) = \sin(x + \frac{\pi}{2}) \)**.