Functions of the form \(y=\cos (k\theta)\)
We now consider cosine functions of the form \(y = \cos k\theta\) and the effects of parameter \(k\).
Optional Investigation: The effects of \(k\) on a cosine graph
- Complete the following table for \(y_1 = \cos \theta\) for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\):
θ \(-\text{360}\)\(\text{°}\) \(-\text{300}\)\(\text{°}\) \(-\text{240}\)\(\text{°}\) \(-\text{180}\)\(\text{°}\) \(-\text{120}\)\(\text{°}\) \(-\text{60}\)\(\text{°}\) \(\text{0}\)\(\text{°}\) \(\cos \theta\) θ \(\text{60}\)\(\text{°}\) \(\text{120}\)\(\text{°}\) \(\text{180}\)\(\text{°}\) \(\text{240}\)\(\text{°}\) \(\text{300}\)\(\text{°}\) \(\text{360}\)\(\text{°}\) \(\cos \theta\) Use the table of values to plot the graph of \(y_1 = \cos \theta\) for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\).
On the same system of axes, plot the following graphs:
- \(y_2 = \cos (-\theta)\)
- \(y_3 = \cos 3\theta\)
- \(y_4 = \cos \cfrac{3\theta}{4}\)
Use your sketches of the functions above to complete the following table:
\(y_1\) \(y_2\) \(y_3\) \(y_4\) period amplitude domain range maximum turning points minimum turning points \(y\)-intercept(s) \(x\)-intercept(s) effect of \(k\) What do you notice about \(y_1 = \cos \theta\) and \(y_2 = \cos (-\theta)\)?
Is \(\cos (-\theta) = -\cos \theta\) a true statement? Explain your answer.
- Can you deduce a formula for determining the period of \(y = \cos k\theta\)?
The effect of the parameter \(k\) on \(y = \cos k\theta\)
The value of \(k\) affects the period of the cosine function.
For \(k > 0\):
For \(k > 1\), the period of the cosine function decreases.
For \(0 < k < 1\), the period of the cosine function increases.
For \(k < 0\):
For \(-1 < k < 0\), the period increases.
For \(k < -1\), the period decreases.
Negative angles: \[\cos (-\theta) = \cos \theta\] Notice that for negative values of \(\theta\), the graph is not reflected about the \(x\)-axis.
Calculating the period:
To determine the period of \(y = \cos k\theta\) we use, \[\text{Period } = \cfrac{\text{360}\text{°}}{|k|}\] where \(|k|\) is the absolute value of \(k\).
\(0 < k < 1\) | \(-1 < k < 0\) |
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\(k > 1\) | \(k < -1\) |
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Example
Question
- Sketch the following functions on the same set of axes for \(-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}\).
- \(y_1 = \cos \theta\)
- \(y_2 = \cos \cfrac{\theta}{2}\)
For each function determine the following:
- Period
- Amplitude
- Domain and range
- \(x\)- and \(y\)-intercepts
- Maximum and minimum turning points
Examine the equations of the form \(y = \cos k\theta\)
Notice that for \(y_2 = \cos \cfrac{\theta}{2}\), \(k < 1\) therefore the period of the graph increases.
Complete a table of values
| θ | \(-\text{180}\)\(\text{°}\) | \(-\text{135}\)\(\text{°}\) | \(-\text{90}\)\(\text{°}\) | \(-\text{45}\)\(\text{°}\) | \(\text{0}\)\(\text{°}\) | \(\text{45}\)\(\text{°}\) | \(\text{90}\)\(\text{°}\) | \(\text{135}\)\(\text{°}\) | \(\text{180}\)\(\text{°}\) |
| \(\cos \theta\) | \(-\text{1}\) | \(-\text{0.71}\) | \(\text{0}\) | \(\text{0.71}\) | \(\text{1}\) | \(\text{0.71}\) | \(\text{0}\) | \(-\text{0.71}\) | \(-\text{1}\) |
| \(\cos \cfrac{\theta}{2}\) | \(\text{0}\) | \(\text{0.38}\) | \(\text{0.71}\) | \(\text{0.92}\) | \(\text{1}\) | \(\text{0.92}\) | \(\text{0.71}\) | \(\text{0.38}\) | \(\text{0}\) |
Sketch the cosine graphs
Complete the table
| \(y_1 = \cos \theta\) | \(y_2 = \cos \cfrac{\theta}{2}\) | |
| period | \(\text{360}\text{°}\) | \(\text{720}\text{°}\) |
| amplitude | \(\text{1}\) | \(\text{1}\) |
| domain | \([-\text{180}\text{°};\text{180}\text{°}]\) | \([-\text{180}\text{°};\text{180}\text{°}]\) |
| range | \([-1;1]\) | \([0;1]\) |
| maximum turning points | \((\text{0}\text{°};1)\) | \((\text{0}\text{°};1)\) |
| minimum turning points | \((-\text{180}\text{°};-1) \text{ and } (\text{180}\text{°};-1)\) | none |
| \(y\)-intercept(s) | \((\text{0}\text{°};1)\) | \((\text{0}\text{°};1)\) |
| \(x\)-intercept(s) | \((-\text{90}\text{°};0) \text{ and } (\text{90}\text{°};0)\) | \((-\text{180}\text{°};0) \text{ and } (\text{180}\text{°};0)\) |
Discovering the characteristics
For functions of the general form: \(f(\theta) = y =\cos k\theta\):
Domain and range
The domain is \(\{ \theta: \theta \in \mathbb{R} \}\) because there is no value for \(\theta\) for which \(f(\theta)\) is undefined.
The range is \(\{ f(\theta): -1 \leq f(\theta) \leq 1, f(\theta) \in \mathbb{R} \}\) or \([-1;1]\).
Intercepts
The \(x\)-intercepts are determined by letting \(f(\theta) = 0\) and solving for \(\theta\).
The \(y\)-intercept is calculated by letting \(\theta = \text{0}\text{°}\) and solving for \(f(\theta)\). \begin{align*} y &= \cos k\theta \\ &= \cos \text{0}\text{°} \\ &= 1 \end{align*} This gives the point \((\text{0}\text{°};1)\).