# How do you find the important points to graph f(x)=cos(x+pi/2)?

May 18, 2016

#### Explanation:

The function $f \left(x\right) = \cos x$ has limited range between $- 1$ and $1$. Further its value is $0$ at $\left(2 n + 1\right) \frac{\pi}{2}$, where $n$ is an integer and its value is $1$ at $x = 2 n \pi$ and is $- 1$ at $\left(2 n + 1\right) \pi$. The urve appears as follows

graph{cosx [-10, 10, -5, 5]}

But we have to draw the graph of $f \left(x\right) = \cos \left(x + \frac{\pi}{2}\right)$. As cosine of any $x$ is always between $- 1$ and $1$, value of $\cos \left(x + \frac{\pi}{2}\right)$ will also be between $- 1$ and $1$.

The only difference that happens is that instead of $x = 0$, as we have $f \left(x\right) = \cos \left(x + \frac{\pi}{2}\right)$ it is at $- \frac{\pi}{2}$ that $f \left(x\right) = 1$. In fact, the entire graph of $\cos x$ will shift by $\frac{\pi}{2}$ towards left.

This is called as phase shift.

and $f \left(x\right)$ will be $0$ at $x = n \pi$, will be $1$ at $x = \left(4 n - 1\right) \frac{\pi}{2}$ and will be $- 1$ at $x = \left(4 n + 1\right) \frac{\pi}{2}$.

The graph appears as shown below.

graph{cos(x+pi/2) [-10, 10, -5, 5]}