Question

Verify the identity sin (x + ???? / 2) = cos(x) for all real numbers x by using a graph.

Answer 1

Answer:

See proof and explanation below.

Step-by-step explanation:

First we can proof this analitically first using the following property:

$sin(a+b) = sin(a) cos(b) + sin (b) cos(a)$

If we apply this into our formula we got:

$sin (x + \frac{\pi}{2}) = sin (x) cos(\frac{\pi}{2}) + sin (\frac{\pi}{2}) cos (x)$

And if we simplify we got:

$sin (x + \frac{\pi}{2}) = sin (\frac{\pi}{2}) cos (x)= cos (x)$

And that complete the proof.

If we analyze the graphs sin(x) and cos (x) we see that we have a gap between two graphs of $\pi/2$ as we can see on the figure attached.

When we do the transformation $sin(x +\pi/2)$ we are moving to the left $\pi/2$ units and then would be exactly the cos function.