A cosine function is a periodic function defined by the x-coordinates of the points on the unit circle in relation to the measure of the circle’s angles (in radians).
It is important to define certain terms when discussing the cosine function. Other related concepts can also be reviewed.
The rule of the basic cosine function is |f(x)= \cos x.|
The following example demonstrates the relationship between the unit circle and the basic cosine function. Each value of |\color{#333FB1}\theta| (in radians) is associated with a point on the circle. By examining the |\color{#EC0000}x|-coordinates of the points, it is possible to plot the graph.
In the animation, move the cursor or point on the curve to observe the link between them.
-
Since the circle can be rotated an infinite number of times, the domain of the function corresponds to the |\mathbb{R}| set.
-
The cosine function has a zero when the angle |\theta| makes a quarter turn |\left(\theta=\dfrac{\pi}{2}\right),| then another when |\theta| completes three quarters of the turn |\left(\theta=\dfrac{3\pi}{2}\right).|
Since the rotation of the circle can occur an infinite number of times, the function has an infinite number of zeros.
||\theta \in\left\{\dots,\dfrac{\pi}{2},\, \dfrac{3\pi}{2},\, \dfrac{5\pi}{2}, \dots\right\}|| -
The maximum value is |1| and the minimum value is |-1,| because the radius of the trigonometric circle is |1| unit. The values occur each time the circle makes a half turn.
||\theta \in\left\{\dots,0,\pi,2\pi,3\pi\dots\right\}||
By analysing the following example, notice that the basic cosine function is obtained by a horizontal shift of |\dfrac{\pi}{2}| units with respect to the basic sine function. In other words, move the |\cos x| function |\dfrac{\pi}{2}| units to the right to obtain the |\sin x| function.
Thus, the following equality is found.
||\begin{align}\sin x&=\cos\left(x-\color{#C58AE1}h\right)\\\sin x&=\cos\left(x-\color{#C58AE1}{\dfrac{\pi}{2}}\right)\end{align}||
The same equality is used when working with trigonometric identities.
Conversely, the following equality can also be determined.
||\cos x=\sin\left(x+\color{#C58AE1}{\dfrac{\pi}{2}}\right)||
In the animation, move the cursor to observe the phase shift between the sine and cosine functions.
The rule of the transformed cosine function is |f(x)=a \cos\big(b(x-h)\big)+k.|
Parameter |a| is related to the amplitude.
Parameter |b| is related to the period.
Parameter |h| is related to the phase shift.
Parameter |k| is related to the axis of oscillation.
The axis of oscillation (also called the midline) corresponds to the horizontal line intersecting the middle of the function.
The axis of oscillation of a cosine function is determined by the parameter |k.|||\color{#3a9a38}{\text{Axis of oscillation}: y = k}||
The axis of oscillation can also be determined using the extrema. ||k=\dfrac{\max+\min}{2}||
The axis of oscillation is needed to properly define the amplitude of a cosine function.
The amplitude |\color{#fa7921}{(A)}| of a cosine function corresponds to the vertical distance between the axis of oscillation and the maximum or minimum.
The amplitude is determined from parameter |a.|||\color{#fa7921}{A=\vert a\vert}||The amplitude can also be determined using the extrema. ||\color{#fa7921}A=\dfrac{\max-\min}{2}||
To find the period, first identify a cycle. In the cosine function, we generally choose a cycle which begins and ends at the extremum. It helps in graphing the cosine function and finding its rule.
The period |(\color{#333fb1}p)| corresponds to the difference between the |x|-values at the beginning and end of a cycle.
The period is determined using parameter |b.|||\color{#333fb1}{p=\dfrac{2\pi}{\vert b\vert}}||
The phase shift corresponds to a horizontal displacement of the transformed cosine function with respect to the basic function.
You can determine the phase shift of a cosine function using parameter |h.|||\color{#C58AE1}{\text{Phase shift}=h}||
Since the cosine function is periodic, there are several possible phase shifts for the same transformed function.
The basic cosine function has been graphed in black in the following image. The transformed cosine function (in purple) may have undergone a phase shift of |\dfrac{\pi}{2},| or a displacement of |\dfrac{\pi}{2}| units to the right in relation to the basic function.
It could also be determined that the function underwent a phase shift of |-\dfrac{3\pi}{2}| units, or a displacement of |\dfrac{3\pi}{2}| units to the left, or even of |\dfrac{5\pi}{2}| units to the right. All three options, and more, are possible and still yield the same transformed cosine function.
