The graphs of
sine and cosine are the same when sine is shifted
left by .
Such a shifting is referred to as a horizontal
shift.


horizontal shift and phase shift: 
If the horizontal shift is positive, the shifting moves to the right. If the horizontal shift is negative, the shifting moves to the left.
The easiest way to determine horizontal shift is to determine by how many units the "starting point" (0,0) of a standard sine curve, y = sin(x), has moved to the right or left. 


Be careful here!! When
trying to determine the direction of a horizontal shift, be sure you have the
function in the proper form,
. (Notice
the
subtraction
of C.)
The horizontal shift is determined by the value of
C.
This expression is really where the value of C is negative and the shift is left.  
In this expression the value of C is positive and the shift is right. 
Another BEWARE!!!
What is the horizontal shift of ? 
At first glance, it may seem that the horizontal shift is
, but it is
NOT. The horizontal shift is actually
, because the
equation is actually 
Physicists and engineers say that the "phase shift" in the above problem
is actually
.
They make a distinction between y = Asin(B(x
 C)) + D and y = Asin(Bx
 C) + D,
calling the phase shift C in the first equation when B =
1, otherwise calling it C in the second equation.
While this distinction exists for physicists and engineers, some mathematics
textbooks use the terms "horizontal shift" and "phase shift" to mean the
same thing. Refer to your textbook, or your instructor, to see what
definition they wish you to use for "phase shift", if they are using that
term. We will be discussing "horizontal shifts".
Horizontal shifts can be applied to all trigonometric functions.
