To solve this problem, we need to understand how the graph of the function [tex]\( y = \cos(x + \frac{\pi}{2}) \)[/tex] compares to the graph of the basic cosine function [tex]\( y = \cos(x) \)[/tex].
We can start by looking at the given function and rewriting it using a known trigonometric identity. The function provided is:
[tex]\[ y = \cos\left(x + \frac{\pi}{2}\right) \][/tex]
A known trigonometric identity tells us that:
[tex]\[ \cos\left(x + \frac{\pi}{2}\right) = -\sin(x) \][/tex]
However, for the purpose of shifting graphs, it's more relevant to consider how the added term inside the cosine function affects the graph. When you add a constant [tex]\( \frac{\pi}{2} \)[/tex] inside the argument of the cosine function, it results in a horizontal shift of the graph. Specifically, adding [tex]\(\frac{\pi}{2}\)[/tex] to the input [tex]\( x \)[/tex] will shift the graph to the left by [tex]\(\frac{\pi}{2}\)[/tex].
To visualize this:
1. The basic graph of [tex]\( y = \cos(x) \)[/tex] has a period of [tex]\( 2\pi \)[/tex] and starts at [tex]\( (0, 1) \)[/tex].
2. By adding [tex]\(\frac{\pi}{2}\)[/tex] to the [tex]\( x \)[/tex] value, every point on the graph of [tex]\( y = \cos(x) \)[/tex] is moved to the left [tex]\(\frac{\pi}{2}\)[/tex] units.
Thus, the transformation [tex]\( y = \cos\left(x + \frac{\pi}{2}\right) \)[/tex] corresponds to shifting the graph of [tex]\( y = \cos(x) \)[/tex]:
[tex]\(\frac{\pi}{2}\)[/tex] units to the left.
