Which of the following sine functions is equivalent to $$ f(x)=\cos (x) $$ ? Option \#1: $$ f(x)=\sin \left(x-\frac{\pi}{2}\right) $$ Option \#2: $$ f(x)=\sin \left(x+\frac{\pi}{2}\right) $$ Option \#3: $$ f(x)=\sin (x+\pi) $$ (1 point) Option \#

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To determine which sine function is equivalent to $$ f(x) = \cos(x) $$, we can use the sine and cosine relationship.

Recall that:
$$
\cos(x) = \sin\left(x + \frac{\pi}{2}\right)
$$
This means that if we shift the sine function by $$ \frac{\pi}{2} $$ to the left, we will get the cosine function.

Now, let's analyze the options:

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

Using the sine angle subtraction identity:
   $$
   \sin\left(x - \frac{\pi}{2}\right) = -\cos(x)
   $$
   This is not equivalent to $$ \cos(x) $$.

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

Using the sine angle addition identity:
   $$
   \sin\left(x + \frac{\pi}{2}\right) = \cos(x)
   $$
   This is equivalent to $$ \cos(x) $$.

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

Using the sine angle addition identity:
   $$
   \sin(x + \pi) = -\sin(x)
   $$
   This is not equivalent to $$ \cos(x) $$.

Thus, the correct option that is equivalent to $$ f(x) = \cos(x) $$ is:

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

Which of the following sine functions is equivalent to \( f(x)=\cos (x) \) ? Option \#1: \( f(x)=\sin \left(x-\frac{\pi}{2}\right) \) Option \#2: \( f(x)=\sin \left(x+\frac{\pi}{2}\right) \) Option \#3: \( f(x)=\sin (x+\pi) \) (1 point) Option \#

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