# Replicating a cosine graph with sine, given transformations?

Say I have a cos function, $y = 1.2cos(0.503x) + 5$ And say I want to replicate it using sine, so expressing the above function in sine so that it gives "the same wave".

How would I do this? I know I have to find a phase shift, but I don't get the steps on how to do it. Also, please no use of Wolframalpha or the like, because I'll have to do these types of problems in class without them.

• do you just want $1.2\cos(0.503x + \pi/2) + 5 = 1.2\sin(0.503(x + \pi/(2*.503)) + 5?$
– abel
Jun 7, 2015 at 2:34
• Not sure what you mean... Jun 7, 2015 at 2:36
• Yes, that is what I want. Jun 7, 2015 at 2:40

Hint:

$\cos(x)$ is $\sin(x)$ shifted by $\pi/2$

$$\therefore \sin(x) = \cos(x - \pi/2)$$

• I see. So heres what I got from that: Since period is 2p/|k| = 2pi / 0.5 which is approx 4pi, and there is a (pi/2) shift between cos and sin, (pi/2) is a fourth of the period 4pi that I have. Therefore, I'd be shifting approx. 12.5/4, and my equation would be 1.2sin(0.503(x+12.5/4)) + 5? Jun 7, 2015 at 2:31
• @user164403: Try graphing it on WolframAlpha to verify.
– user174622
Jun 7, 2015 at 2:34
• They line up and do in fact work. Thank you for your help. Jun 7, 2015 at 2:36
• @user164403 Then you're done.
– user174622
Jun 7, 2015 at 2:41