# Transformed Cosine & Sine Curves - Wave Function

## (Working with Degrees)

Transformed cosine and sine curves, sometimes called wave functions, are cosine and sine curves on which we have carried-out a series of transformations.

In their most general form, wave functions are defined by the equations: $y = a.cos\begin{pmatrix}b(x-c)\end{pmatrix}+d$

and
$y = a.sin\begin{pmatrix}b(x-c)\end{pmatrix}+d$ Where:
• $$a$$ is known as the amplitude
• $$b$$ is known as the wave number, also called the angular frequency
• $$c$$ is known as the phase shift
• $$d$$ is known as the vertical shift or rest position.
In this section we define and learn how to find each of these when given a cosine or sine curve.

By the end of this, we should be quite comfortable with any cosine or sine curve looking like the one shown here:

## The Amplitude: $$a$$

The amplitude of a cosine, or a sine, curve is the amount by which the curve oscillates (either above or below) its "rest", or mean, value.

The amplitude is illustrated here. The pink curve is: $y = 3.cos(x)$ The dotted curve is the "regular" $$y = cos(x)$$ curve.

We can see that:

• the pink curve $$y=3.cos(x)$$ oscillates $$3$$ units either side of the $$x$$-axis, it has an amplitude of $$3$$: $$a = 3$$.
• the dotted curve $$y=cos(x)$$ oscillates $$1$$ unit either side of the $$x$$-axis, it has an amplitude of $$1$$: $$a = 1$$.
At times it won't be as easy to "simply see" the amplitude and we'll need a formula to find it

## Formula for amplitude $$a$$

Given the graph of a cosine, or sine, curve we can find the value of its amplitude using the formula: $a = \frac{y_{\text{max}} - y_{\text{min}}}{2}$ Where:

• $$y_{\text{min}}$$ is the lowest point on the curve.
• $$y_{\text{max}}$$ is the highest point on the curve.

## The Wave Number: $$b$$

Given the graph of either a cosine or a sine function, the wave number $$b$$, also known as angular frequency, tells us:

how many fully cycles the curve does every $$360^{\circ}$$ interval

It is inversely proportional to the function's period $$T$$. This means that the greater $$b$$ is: the smaller the period becomes.

The wave number $$b$$ is illustrated here, using the sine function defined by: $y = 2.sin(3x)$

We can see from the equation $$y = 2.sin(3x)$$ that the wave number is: $b = 3$ On the graph, this can be seen from the fact that the curve completes $$3$$ full cycles every $$360^{\circ}$$.

We can also find the wave number if we know a cosine, or sine, curve's period $$T$$. In that case, the wave number $$b$$ can be found using the formula given below.

## Formula: wave number $$b$$

Given a cosine, or sine, curve with period $$T$$, its wave number $$b$$ can be calculated using the formula: $b = \frac{360}{T}$

## Example

Consider the cosine curve shown here:

Given this curve has equation $$y = 3.cos(bx)$$, find the value of $$b$$.

### Solution

We can see that:

• the curve completes $$2$$ cycles every $$360^{\circ}$$
• it has a period $$T = 180^{\circ}$$.
The first bullet point, above, tells us that this curve's wave number is equal to $$2$$, in other words: $b = 2$ Alternatively, we can use the fact that this curve's period is $$T = 180^{\circ}$$ and use our formula: \begin{aligned} b & = \frac{360}{T} \\ & = \frac{360}{180} \\ b & = 2 \end{aligned}

## Finding the Phase Shift: $$c$$

The phase shift correponds to a horizontal translation of the cosine, or sine, curve.

Graphically, it can be seen by looking at how far to the right (or to the left) the curve without $$c$$ has been moved.

For example, the following graph shows the curve: $y = 3.sin\begin{pmatrix}x - 30 \end{pmatrix}$ as well as the curve $$y = 3.sin(x)$$ in dotted gray lines.

We can see that the phase shift of $$30^{\circ}$$ moves the entire curve $$y=3.sin(x)$$ $$30^{\circ}$$ to the right.

## Method

For a cosine curve, $$y = a.cos\begin{pmatrix}b(x-c)\end{pmatrix} + d$$, or a sine curve, $$y = a.sin\begin{pmatrix}b(x-c)\end{pmatrix} + d$$, we can find the phase shift $$c$$ directly from its curve as follows:

• For a cosine curve $$y = a.cos\begin{pmatrix}b(x-c)\end{pmatrix} + d$$

## Example

The following curve has equation $$y = a.cos\begin{pmatrix}b(x - c) \end{pmatrix}$$.

1. Find:
1. The value of the amplitude $$a$$.
2. The value of the wave number $$b$$.
3. The value of the phase shift $$c$$.
2. Given that this curve's equation can be written in the form: $y = a.cos\begin{pmatrix}bx+p\end{pmatrix}$ state the value of $$p$$.

### Solution

1. To find the amplitude we can use the formula we saw further-up: $a = \frac{y_{\text{max}} - y_{\text{min}}}{2}$ with:
• $$y_{\text{max}} = 4$$
• $$y_{\text{min}} = -4$$
So the amplitude is: \begin{aligned} a & = \frac{y_{\text{max}} - y_{\text{min}}}{2}\\ & = \frac{4 - (-4)}{2}\\ & = \frac{4+4}{2}\\ & = \frac{8}{2}\\ a & = 4 \end{aligned} At this stage we can state that this curve has an equation of the form: $y = 4.cos\begin{pmatrix}b(x - c) \end{pmatrix}$
2. To find the wave number $$b$$, we use our formula: $b = \frac{360}{T}$ Where the period $$T$$ can be read off the graph.
Indeed we can see that: $T = 180^{\circ}$ So we find: \begin{aligned} b &= \frac{360}{T}\\ & = \frac{360}{180} \\ b& = 2 \end{aligned} At this stage we can state that this curve has an equation of the form: $y = 4.cos\begin{pmatrix}2(x - c) \end{pmatrix}$
3. To find the phase shift $$c$$, we can compare the curve we're given to the curve without any phase shift, that's: $y = 4.cos\begin{pmatrix}2x \end{pmatrix}$ The phase shift will then equal to the amount by which the curve without the phase shift (shown in dotted gray below) would have to be horizontally translated to reach the curve we were initially given.

## The Vertical Shift: $$d$$

### (the "rest" position)

Given the graph of either a cosine or a sine function, to find the rest position $$d$$, which is the amount the curve has been translated vertically, we can use the following formula.

This is illustrated here:

• the pink curve shows $$y = 3.sin(x)+2$$
• the dotted curve shows $$y = 3.sin(x)$$

We can see that the pink curve, $$y=3.sin(x)+2$$ is the same as the dotted curve but it has been moved $$2$$ units upwards.

The pink curve $$y=3.sin(x)+2$$ oscillated above and below the "new" rest poistion $$y=2$$.

If we can't see/read the value of the vertical shift directly off the graph, we can use the following formula to find the value of $$d$$.

## Formula: vertical translation $$d$$

$d = \frac{y_{\text{max}} + y_{\text{min}}}{2}$ Where:

• $$y_{\text{min}}$$ is the lowest point on the curve.
• $$y_{\text{max}}$$ is the highest point on the curve.

## Example

The curve shown here has equation: $y = a.sin(bx)+d$

1. Find the value of $$a$$.
2. Find the value of $$b$$.
3. Find the value of $$d$$.

### Solution

1. The amplitude $$a$$ can be found using our formula: $a = \frac{y_{\text{max}} - y_{\text{min}}}{2}$ Looking at the curve:
We can see that:
• $$y_{\text{max}} = 1$$
• $$y_{\text{min}} = -7$$
So we find: \begin{aligned} a &= \frac{y_{\text{max}} - y_{\text{min}}}{2}\\ & = \frac{1 - (-7)}{2} \\ & = \frac{1+7}{2} \\ & = \frac{8}{2}\\ a & = 4 \end{aligned} $y = 4.sin(bx)+d$
2. The wave number $$b$$ can be found using the curve's period $$T$$ and the formula: $b = \frac{360}{T}$ Looking at the curve we can find its period:
We can see that its period is $$T = 180^{\circ}$$, so the formula leads to: \begin{aligned} b &= \frac{360}{T} \\ & = \frac{360}{180} \\ b & = 2 \end{aligned} At this stage we can state that the curve has an equation of the form: $y = 4.sin(2x)+d$
3. To find the vertical shift $$d$$ we can use our formula: $d = \frac{y_{\text{max}} + y_{\text{min}}}{2}$ Looking at the curve we can see the maximum and minimum $$y$$-values:
Indeed we can see that:
• $$y_{\text{max}} = 1$$
• $$y_{\text{min}} = -7$$
So our formula becomes: \begin{aligned} d &= \frac{y_{\text{max}} + y_{\text{min}}}{2}\\ & = \frac{1+(-7)}{2} \\ & = \frac{1-7}{2}\\ & = \frac{-6}{2} \\ d & = -3 \end{aligned} Finally we can state this curve's equation: $y = 4.sin(2x)-3$

## Exercise 1

The curve, shown below, has equation: $y = a.sin(bx)$

1. Find the value of $$a$$.
2. Find the value of $$b$$.

1. The amplitude is $$a = 2$$.
2. The wave number is $$b=3$$.
This curve's equation is therefore: $y = 2.sin(3x)$

## Exercise 2

The curve, shown below, has equation: $y = a.cos\begin{pmatrix}x-c\end{pmatrix}$

1. Find the value of $$a$$.
2. Find the value of $$c$$.

1. The amplitude is $$a = 4$$.
2. The phase shift is $$c = 30^{\circ}$$.
This curve's equation is therefore: $y = 4.cos(x - 30)$

## Exercise 3

The curve, shown below, has equation: $y = a.cos\begin{pmatrix}b(x-c)\end{pmatrix}$

1. Find the value of $$a$$.
2. State this function's period.
3. Find the value of $$b$$.
4. Find the value of $$c$$.

1. The amplitude is $$a = 3$$.
2. The period is $$T = 180^{\circ}$$.
3. The wave number is: \begin{aligned} b &= \frac{360}{T} \\ & = \frac{360}{180} \\ b & = 2 \end{aligned}
4. The phase shift is $$c = 60^{\circ}$$.
The curve's equation is $$y = 3.cos\begin{pmatrix} 2(x-60)\end{pmatrix}$$