# How do you graph sin(2x-pi/2)?

Feb 7, 2017

= 2cos (2x)

#### Explanation:

Use trig unit circle and property of complement arcs:
y = - 2sin (2x - pi/2) = 2cos (2x)

Feb 7, 2017

See below for step-by-step analysis and graphing

#### Explanation:

Let's start with the standard $\textcolor{b l a c k}{\sin \left(x\right)}$ function
and in particular let's note the basic cycle of the $\sin \left(x\right)$ function:

Notice that the argument range for the basic cycle is $\left[0 , 2 \pi\right]$

$\textcolor{m a \ge n t a}{\text{<><><><><><><><><><><><><><><><><><><><><><><><><><><> }}$

Now let's consider what happens with $\sin \left(x - \frac{\pi}{2}\right)$
The basic cycle still requires an argument range of $\left[0 , 2 \pi\right]$,
that is $\left(x - \frac{\pi}{2}\right) \in \left[0 , 2 \pi\right]$
but for this to be possible, $x \in \left[\frac{\pi}{2} , \frac{5 \pi}{2}\right]$

That is the basic cycle will appear to have shifted to the right by $\frac{\pi}{2}$

$\textcolor{m a \ge n t a}{\text{<><><><><><><><><><><><><><><><><><><><><><><><><><><> }}$

Next let's consider what happens when we double the value of the variable $x$ causing the argument expression to become $\left(2 x - \frac{\pi}{2}\right)$
The basic cycle still needs an argument range $\left[0 , 2 \pi\right]$,
that is $\left(2 x - \frac{\pi}{2}\right) \in \left[0 , 2 \pi\right]$
which, as we have seen, implies $2 x \in \left[\frac{\pi}{2} , \frac{5 \pi}{2}\right]$
which, to be possible, means $x \in \left[\frac{\pi}{4} , \frac{5 \pi}{4}\right]$

The horizontal distance from the origin on the X-axis seems to have been "squeezed" by a factor of $\frac{1}{2}$

$\textcolor{m a \ge n t a}{\text{<><><><><><><><><><><><><><><><><><><><><><><><><><><> }}$

Finally, what happens to our basic cycle when we multiply the entire expression by $\left(- 2\right)$ to get $y = - 2 \sin \left(2 x - \frac{\pi}{2}\right)$?
The value of each y-coordinate is scaled by a factor of $2$
(you might think of this as the coordinate is reflected in the X-axis and then stretched by $2$ away form the X-axis).

$\textcolor{m a \ge n t a}{\text{<><><><><><><><><><><><><><><><><><><><><><><><><><><> }}$

Of course, this only gives us the basic cycle.

For the full graph this cycle is repeated infinitely in both directions: