There are several different ways to express "formulas" or equations that relate the sine and cosine functions. To the student new to trigonometry they form a bewildering array of nonsense formulas to be memorized. (Memorization of nonsense is distasteful, utterly boring, and destined to trip one up and make one feel like an idiot when the memorized nonsense is misapplied.) Fortunately there is a simple, reliable "trick" or method whereby with a quick rotation about the unit circle, one can find the appropriate shift to convert a sine function into a cosine function, or vice versa.
This method is illustrated for every problem below on the solutions page. However, it is a good idea to at least try to figure out a method for yourself first before looking at the method I made up. (Note: pi may appear as the letter "p" in older Netscape browsers in the document below. Newer Netscape browsers and Explorer browsers should show the Greek letter pi correctly.)
cos(x + p)
=
cos(x - p)
=
cos(x + 
) =
cos (x - 
) =
sin (x + 
) =
sin (x - 
) =
cos(x - 
) =
cos(x + 
) =
sin (x + p)
=
sin(x - p)
=