Replicating cosine/sine graph, but with reflections?

Question

Replicating a cosine graph with sine, given transformations?

This is an extension question of my previous post.

Say I have an equation like $y = 7 \cos(0.96(x-3)) + 11$.

How would I find the sine equivalent that lines exactly with it? I thought that $\sin$ and $\cos$ differ only by a phase shift of $\displaystyle -\frac{\pi}{2}$, when do I need to use reflections?

Thanks

Answer 1

what about $$\begin{align} y & = 7 \cos(0.96(x-3)) + 11 \\ &= 7 \sin(\pi/2 + 0.96(x-3)) + 11\\ & = 7\sin(0.96(x - 3 + \pi/(2 \times 0.96)) + 11\\ &= 7 \sin(0.96(x- 1.3637) + 11? \end{align}$$

Answer 2

$A * \sin( B * x - C ) + D$

A = amplitude

B = Period, usually given as pi/B

C = phase shift (or horizontal offset if you prefer)

D = vertical offset

Since you are only changing the horizontal to correctly duplicate sine in cosine or vice versa, the ONLY thing that should change is the phase shift, which is $\pm \pi/2$ to cause it to align with the opposite function.

addendum A & B are assumed to be 1 if not otherwise given C & D are assumed to be 0 if not otherwise given