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### Phase Shift in Sine Function

```
Date: 02/19/99 at 22:37:08
From: Corine Clark
Subject: Phase shift in sine function

I have a trigonometry test on Monday, and I do not understand phase
shift in functions. The problem I am stuck on is y = sin(4x+pi/3). I
have figured out that the amplitude is 1, the period is 90 degrees,
and the phase shift is -15 degrees. I know how to graph this function
without the shift, and I know the graph shifts to the left. What I
do not know is how to move the function over correctly. Could you
please show me an example of a shifted graph?

Thanks for your time. I appreciate it!
```

```
Date: 02/20/99 at 15:31:27
From: Doctor Rick
Subject: Re: Phase shift in sine function

*                               *
*  B  *                         *
*         *                     *
*           *                   *
*             *                 *
----*---------------*---------------*---------
*A               C*             *E
*                   *           *
*                     *         *
*     *
*
D

Okay, here is an attempt at drawing a sine curve. I labeled the points
A, B, C, D, and E. These are points on the curve that you get when the
argument of the sine function (what's inside the parentheses) is 0,
pi/2, pi, 3pi/2, and 2pi.

Your function is y = sin(4x+pi/3). What will x be at point A? You want

4x+pi/3 = 0

so that y = sin(0) = 0. You can solve for this and you will find
x = -pi/12 (which is -15 degrees, but you should leave it as it is).
What will x be at point C? You want

4x+pi/3 = pi

so that y = sin(pi) = 0 again. You can solve this to get x = pi/6.
Since this is twice pi/12, the y axis is 1/3 of the way between
A and C.

|  *                               *
*  B  *                         *
* |       *                     *
*  |        *                   *
*   |         *                 *
----*----+----------*---------------*---------
*|    |          |*             *E
*-pi   |         pi *           *
*  --   |         --  *         *
12   |          6    *     *
|                  *
|                  D

Another way of seeing this is to note that when x = 0 you are
evaluating sin(pi/3), and pi/3 is 60 degrees, which falls where you see
it on the sine curve: 2/3 of the way up the curve toward B.

You can work out the other points if you need to. This is a way to be
sure you've got it right. You can see now that you had the information
you needed: the sine curve is shifted left by pi/12. Perhaps the only
thing that was confusing you was that you converted this shift to
degrees.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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