Discovering the Characteristics of Cosine Functions
Discovering the Characteristics of Cosine Functions
Domain and range
For \(f(\theta)=a\cos\theta +q\), the domain is \([0°;360°]\)
It is easy to see that the range of \(f(\theta)\) will be the same as the range of \(a\sin\theta +q\). This is because the maximum and minimum values of \(a\cos \theta + q\) will be the same as the maximum and minimum values of \(a\sin\theta +q\).
For \(a>0\) the range of \(f(\theta)=a\cos\theta +q\) is \(\{f(\theta):f(\theta)\in [-a+q;a+q]\}\)
For \(a<0\) the range of \(f(\theta)=a \cos\theta +q\) is \(\{f(\theta):f(\theta)\in [a+q;-a+q]\}\)
Period
The period of \(y=a\cos\theta +q\) is \(360°\). This means that one cosine wave is completed in \(360°\).
Intercepts
The \(y\)-intercept of \(f(\theta)=a\cos\theta +q\) is calculated in the same way as for sine.
\begin{align*} y& = f(0°) \\ & = a\cos 0°+q \\ & = a(1)+q \\ & = a+q \end{align*}This gives the point \((0°;a+q)\).
Example
Question
Sketch the graph of \(f(\theta)=2\cos\theta +3\) for \(\theta \in [0°;360°]\).
Examine the standard form of the equation
From the equation we see that \(a>1\) so the graph is stretched vertically. We also see that \(q>0\) so the graph is shifted vertically upwards by \(\text{3}\) units.
Substitute values for \(\theta\)
|
\(\theta\) |
\(0°\) |
\(30°\) |
\(60°\) |
\(90°\) |
\(120°\) |
\(150°\) |
\(180°\) |
\(210°\) |
\(240°\) |
\(270°\) |
\(300°\) |
\(330°\) |
\(360°\) |
|
\(f(\theta)\) |
\(\text{5}\) |
\(\text{4.73}\) |
\(\text{4}\) |
\(\text{3}\) |
\(\text{2}\) |
\(\text{1.27}\) |
\(\text{1}\) |
\(\text{1.27}\) |
\(\text{2}\) |
\(\text{3}\) |
\(\text{4}\) |
\(\text{4.73}\) |
\(\text{5}\) |
Plot the points and join with a smooth curve
Domain: \([0°;360°]\)
Range: \([1;5]\)
\(x\)-intercepts: none
\(y\)-intercept: \((0°;5)\)
Maximum turning points: \((0°;5)\), \((360°;5)\)
Minimum turning point: \((180°;1)\)
This lesson is part of:
Functions I