Mathematics » Functions II » The Cosine Function

Functions of the Form y = cos (kθ)

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

1. 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 $$\text{60}$$$$\text{°}$$ $$\text{120}$$$$\text{°}$$ $$\text{180}$$$$\text{°}$$ $$\text{240}$$$$\text{°}$$ $$\text{300}$$$$\text{°}$$ $$\text{360}$$$$\text{°}$$ $$\cos \theta$$
2. Use the table of values to plot the graph of $$y_1 = \cos \theta$$ for $$-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}$$.

3. On the same system of axes, plot the following graphs:

1. $$y_2 = \cos (-\theta)$$
2. $$y_3 = \cos 3\theta$$
3. $$y_4 = \cos \cfrac{3\theta}{4}$$
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 $$k$$
5. What do you notice about $$y_1 = \cos \theta$$ and $$y_2 = \cos (-\theta)$$?

6. Is $$\cos (-\theta) = -\cos \theta$$ a true statement? Explain your answer.

7. 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 $$k$$.

 $$0 < k < 1$$ $$-1 < k < 0$$ $$k > 1$$ $$k < -1$$

Example

Question

1. Sketch the following functions on the same set of axes for $$-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}$$.
1. $$y_1 = \cos \theta$$
2. $$y_2 = \cos \cfrac{\theta}{2}$$
2. For each function determine the following:

1. Period
2. Amplitude
3. Domain and range
4. $$x$$- and $$y$$-intercepts
5. 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 $$-\text{180}$$$$\text{°}$$ $$-\text{135}$$$$\text{°}$$ $$-\text{90}$$$$\text{°}$$ $$-\text{45}$$$$\text{°}$$ $$\text{0}$$$$\text{°}$$ $$\text{45}$$$$\text{°}$$ $$\text{90}$$$$\text{°}$$ $$\text{135}$$$$\text{°}$$ $$\text{180}$$$$\text{°}$$ $$\cos \theta$$ $$-\text{1}$$ $$-\text{0.71}$$ $$\text{0}$$ $$\text{0.71}$$ $$\text{1}$$ $$\text{0.71}$$ $$\text{0}$$ $$-\text{0.71}$$ $$-\text{1}$$ $$\cos \cfrac{\theta}{2}$$ $$\text{0}$$ $$\text{0.38}$$ $$\text{0.71}$$ $$\text{0.92}$$ $$\text{1}$$ $$\text{0.92}$$ $$\text{0.71}$$ $$\text{0.38}$$ $$\text{0}$$

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 $$(\text{0}\text{°};1)$$.