# Translating Sine and Cosine Functions

## Key Questions

• For an equation:

A vertical translation is of the form:
$y = \sin \left(\theta\right) + A$ where $A \ne 0$
OR $y = \cos \left(\theta\right) + A$

Example: $y = \sin \left(\theta\right) + 5$ is a $\sin$ graph that has been shifted up by 5 units

The graph $y = \cos \left(\theta\right) - 1$ is a graph of $\cos$ shifted down the y-axis by 1 unit

A horizontal translation is of the form:
$y = \sin \left(\theta + A\right)$ where $A \ne 0$

Examples:
The graph $y = \sin \left(\theta + \frac{\pi}{2}\right)$ is a graph of $\sin$ that has been shifted $\frac{\pi}{2}$ radians to the right

For a graph:
I'm to illustrate with an example given above:

For compare:
$y = \cos \left(\theta\right)$
graph{cosx [-5.325, 6.675, -5.16, 4.84]}

and

$y = \cos \left(\theta\right) - 1$
graph{cosx -1 [-5.325, 6.675, -5.16, 4.84]}
To verify that the graph of $y = \cos \left(\theta\right) - 1$ is a vertical translation, if you look on the graph,

the point where $\theta = 0$ is no more at $y = 1$ it is now at $y = 0$

That is, the original graph of $y = \cos \theta$ has been shifted down by 1 unit.

Another way to look at it is to see that, every point has been brought down 1 unit!

• I think you'll find a useful answer here: http://socratic.org/trigonometry/graphing-trigonometric-functions/translating-sine-and-cosine-functions

Vertical translation

Graphing $y = \sin x + k$ Which is the same as $y = k + \sin x$:

In this case we start with a number (or angle) $x$. We find the sine of $x$, which will be a number between $- 1$ and $1$. The after that, we get $y$ by adding $k$. (Remember that $k$ could be negative.)

This gives us a final $y$ value betwee $- 1 + k$ and $1 + k$.

This will translate the graph up if $k$ is positive ($k > 0$)
or down if $k$ is negative ($k < 0$)

Examples:

$y = \sin x$
graph{y=sinx [-5.578, 5.52, -1.46, 4.09]}

$y = \sin x + 2 = 2 + \sin x$
graph{y=sinx+2 [-5.578, 5.52, -1.46, 4.09]}

$y = \sin x - 4 = - 4 + \sin x$
graph{y=sinx-4 [-5.58, 5.52, -5.17, 0.38]}

The reasoning is the same for $y = \cos x + k = k + \cos x$, but the starting graph looks different, so the final graph is also different:

$y = \cos x$
graph{y=cosx [-5.578, 5.52, -1.46, 4.09]}

$y = \cos x + 2 = 2 + \cos x$
graph{y=cosx+2 [-5.578, 5.52, -1.46, 4.09]}