Mathematics » Functions II » The Cosine Function

# Sketching Cosine Graphs

## Example

### Question

Sketch the graph of $$f(\theta) = \cos (\text{180}\text{°} – 3\theta)$$ for $$\text{0}\text{°} \leq \theta \leq \text{360}\text{°}$$.

### Examine the form of the equation

Write the equation in the form $$y = \cos k(\theta + p)$$. \begin{align*} f(\theta) &= \cos (\text{180}\text{°} – 3\theta)\\ &= \cos (-3\theta + \text{180}\text{°}) \\ &= \cos ( -3(\theta – \text{60}\text{°}) ) \\ &= \cos 3(\theta – \text{60}\text{°}) \end{align*}

To draw a graph of the above function, the standard cosine graph, $$y = \cos \theta$$, must be changed in the following ways:

• decrease the period by a factor of $$\text{3}$$
• shift to the right by $$\text{60}\text{°}$$.

### Complete a table of values

 θ $$\text{0}$$$$\text{°}$$ $$\text{45}$$$$\text{°}$$ $$\text{90}$$$$\text{°}$$ $$\text{135}$$$$\text{°}$$ $$\text{180}$$$$\text{°}$$ $$\text{225}$$$$\text{°}$$ $$\text{270}$$$$\text{°}$$ $$\text{315}$$$$\text{°}$$ $$\text{360}$$$$\text{°}$$ $$f(\theta)$$ $$-\text{1}$$ $$\text{0.71}$$ $$\text{0}$$ $$-\text{0.71}$$ $$\text{1}$$ $$-\text{0.71}$$ $$\text{0}$$ $$\text{0.71}$$ $$-\text{1}$$

### Plot the points and join with a smooth curve Period: $$\text{120}$$$$\text{°}$$

Amplitude: $$\text{1}$$

Domain: $$[\text{0}\text{°};\text{360}\text{°}]$$

Range: $$[-1;1]$$

Maximum turning point: $$(\text{60}\text{°};1)$$, $$(\text{180}\text{°};1)$$ and $$(\text{300}\text{°};1)$$

Minimum turning point: $$(\text{0}\text{°}; -1)$$, $$(\text{120}\text{°};-1)$$, $$(\text{240}\text{°};-1)$$ and $$(\text{360}\text{°};-1)$$

$$y$$-intercepts: $$(\text{0}\text{°};-1)$$

$$x$$-intercept: $$(\text{30}\text{°};0)$$, $$(\text{90}\text{°};0)$$, $$(\text{150}\text{°};0)$$, $$(\text{210}\text{°};0)$$, $$(\text{270}\text{°};0)$$ and $$(\text{330}\text{°};0)$$

## Example

### Question

Given the graph of $$y = a \cos (k\theta + p)$$, determine the values of $$a$$, $$k$$, $$p$$ and the minimum turning point. ### Determine the value of $$k$$

From the sketch we see that the period of the graph is $$\text{360}\text{°}$$, therefore $$k = 1$$.

$y = a \cos ( \theta + p)$

### Determine the value of $$a$$

From the sketch we see that the maximum turning point is $$(\text{45}\text{°};2)$$, so we know that the amplitude of the graph is $$\text{2}$$ and therefore $$a = 2$$.

$y = 2 \cos ( \theta + p)$

### Determine the value of $$p$$

Compare the given graph with the standard cosine function $$y = \cos \theta$$ and notice the difference in the maximum turning points. We see that the given function has been shifted to the right by $$\text{45}$$$$\text{°}$$, therefore $$p = \text{45}\text{°}$$.

$y = 2 \cos ( \theta – \text{45}\text{°})$

### Determine the minimum turning point

At the minimum turning point, $$y = -2$$:

\begin{align*} y &= 2 \cos ( \theta – \text{45}\text{°}) \\ -2 &= 2 \cos ( \theta – \text{45}\text{°}) \\ -1 &= \cos ( \theta – \text{45}\text{°}) \\ \cos^{-1}(-1) &= \theta – \text{45}\text{°} \\ \text{180}\text{°} &= \theta – \text{45}\text{°} \\ \text{225}\text{°} &= \theta \end{align*}

This gives the point $$(\text{225}\text{°};-2)$$.