# How do you graph y = cos(90-A) as sinA?

Sep 17, 2015

Just graphing it. $\cos \left(90 - A\right) = \sin \left(A\right)$

#### Explanation:

There are many ways to go around and show this, the easiest is simply plugging the 5 main values and show that it matches (even though this isn't a proof it's enough to understand)

$\cos \left(90 - 0\right) = \cos \left(90\right) = 0 = \sin \left(0\right)$
$\cos \left(90 - 30\right) = \cos \left(60\right) = \frac{1}{2} = \sin \left(30\right)$
$\cos \left(90 - 45\right) = \cos \left(45\right) = \frac{\sqrt{2}}{2} = \sin \left(45\right)$
$\cos \left(90 - 60\right) = \cos \left(30\right) = \frac{\sqrt{3}}{2} = \sin \left(60\right)$
$\cos \left(90 - 90\right) = \cos \left(0\right) = 1 = \sin \left(90\right)$
$\cos \left(90 - 180\right) = \cos \left(- 90\right) = \cos \left(90\right) = 0 = \sin \left(180\right)$

We can proof this though, by using the property
$\cos \left(a - b\right) = \cos \left(a\right) \cos \left(b\right) + \sin \left(a\right) \sin \left(b\right)$

So

$\cos \left(90 - x\right) = \cos \left(90\right) \cos \left(x\right) + \sin \left(90\right) \sin \left(x\right)$

As we've seen on the list above, $\cos \left(90\right) = 0$ and $\sin \left(90\right) = 1$, thus:

$\cos \left(90 - x\right) = 0 \cos \left(x\right) + 1 \sin \left(x\right)$
$\cos \left(90 - x\right) = \sin \left(x\right)$