Mathematics » Functions II » The Tangent Function

Functions of the Form y = tan (kθ)

Optional Investigation: The effects of $$k$$ on a tangent graph

1. Complete the following table for $$y_1 = \tan \theta$$ for $$-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}$$:
 θ $$-\text{360}$$$$\text{°}$$ $$-\text{300}$$$$\text{°}$$ $$-\text{240}$$$$\text{°}$$ $$-\text{180}$$$$\text{°}$$ $$-\text{120}$$$$\text{°}$$ $$-\text{60}$$$$\text{°}$$ $$\text{0}$$$$\text{°}$$ $$\tan \theta$$ θ $$\text{60}$$$$\text{°}$$ $$\text{120}$$$$\text{°}$$ $$\text{180}$$$$\text{°}$$ $$\text{240}$$$$\text{°}$$ $$\text{300}$$$$\text{°}$$ $$\text{360}$$$$\text{°}$$ $$\tan \theta$$
2. Use the table of values to plot the graph of $$y_1 = \tan \theta$$ for $$-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}$$.

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

1. $$y_2 = \tan (-\theta)$$
2. $$y_3 = \tan 3\theta$$
3. $$y_4 = \tan \cfrac{\theta}{2}$$
4. Use your sketches of the functions above to complete the following table:

 $$y_1$$ $$y_2$$ $$y_3$$ $$y_4$$ period domain range $$y$$-intercept(s) $$x$$-intercept(s) asymptotes effect of $$k$$
5. What do you notice about $$y_1 = \tan \theta$$ and $$y_2 = \tan (-\theta)$$?

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

7. Can you deduce a formula for determining the period of $$y = \tan k\theta$$?

The effect of the parameter on $$y = \tan k\theta$$

The value of $$k$$ affects the period of the tangent function. If $$k$$ is negative, then the graph is reflected about the $$y$$-axis.

• For $$k > 0$$:

For $$k > 1$$, the period of the tangent function decreases.

For $$0 < k < 1$$, the period of the tangent function increases.

• For $$k < 0$$:

For $$-1 < k < 0$$, the graph is reflected about the $$y$$-axis and the period increases.

For $$k < -1$$, the graph is reflected about the $$y$$-axis and the period decreases.

Negative angles: $\tan (-\theta) = -\tan \theta$

Calculating the period:

To determine the period of $$y = \tan k\theta$$ we use, $\text{Period} = \cfrac{\text{180}\text{°}}{|k|}$ where $$|k|$$ is the absolute value of $$k$$.

 $$k > 0$$ $$k < 0$$

Example

Question

1. Sketch the following functions on the same set of axes for $$-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}$$.
1. $$y_1 = \tan \theta$$
2. $$y_2 = \tan \cfrac{3\theta}{2}$$
2. For each function determine the following:

• Period
• Domain and range
• $$x$$- and $$y$$-intercepts
• Asymptotes

Examine the equations of the form $$y = \tan k\theta$$

Notice that $$k > 1$$ for $$y_2 = \tan \cfrac{3\theta}{2}$$, therefore the period of the graph decreases.

Complete a table of values

 θ $$-\text{180}$$$$\text{°}$$ $$-\text{135}$$$$\text{°}$$ $$-\text{90}$$$$\text{°}$$ $$-\text{45}$$$$\text{°}$$ $$\text{0}$$$$\text{°}$$ $$\text{45}$$$$\text{°}$$ $$\text{90}$$$$\text{°}$$ $$\text{135}$$$$\text{°}$$ $$\text{180}$$$$\text{°}$$ $$\tan \theta$$ $$\text{0}$$ $$\text{1}$$ undef $$-\text{1}$$ $$\text{0}$$ $$\text{1}$$ undef $$-\text{1}$$ $$\text{0}$$ $$\tan \cfrac{3\theta}{2}$$ undef $$-\text{0.41}$$ $$\text{1}$$ $$-\text{2.41}$$ $$\text{0}$$ $$\text{2.41}$$ $$-\text{1}$$ $$\text{0.41}$$ undef

Complete the table

 $$y_1 = \tan \theta$$ $$y_2 = \tan \cfrac{3\theta}{2}$$ period $$\text{180}$$$$\text{°}$$ $$\text{120}$$$$\text{°}$$ domain $$\{\theta: -\text{180}\text{°} \leq \theta \leq \text{180}\text{°}, \theta \ne -\text{90}\text{°}; \text{90}\text{°}\}$$ $$\{\theta: -\text{180}\text{°} < \theta < \text{180}\text{°}, \theta \ne -\text{60}\text{°}; \text{60}\text{°}\}$$ range $$\{f(\theta): f(\theta) \in \mathbb{R}\}$$ $$\{f(\theta): f(\theta) \in \mathbb{R}\}$$ $$y$$-intercept(s) $$(\text{0}\text{°};0)$$ $$(\text{0}\text{°};0)$$ $$x$$-intercept(s) $$(-\text{180}\text{°};0)$$, $$(\text{0}\text{°};0)$$ and $$(\text{180}\text{°};0)$$ $$(-\text{120}\text{°};0)$$, $$(\text{0}\text{°};0)$$ and $$(\text{120}\text{°};0)$$ asymptotes $$\theta = -\text{90}\text{°}$$ and $$\theta = \text{90}\text{°}$$ $$\theta = -\text{180}\text{°}$$; $$-\text{60}\text{°}$$ and $$\text{180}\text{°}$$

Discovering the characteristics

For functions of the general form: $$f(\theta) = y =\tan k\theta$$:

Domain and range

The domain of one branch is $$\{ \theta: -\cfrac{\text{90}\text{°}}{k} < \theta < \cfrac{\text{90}\text{°}}{k}, \theta \in \mathbb{R}\}$$ because $$f(\theta)$$ is undefined for $$\theta = -\cfrac{\text{90}\text{°}}{k}$$ and $$\theta = \cfrac{\text{90}\text{°}}{k}$$.

The range is $$\{ f(\theta): f(\theta) \in \mathbb{R} \}$$ or $$(-\infty; \infty)$$.

Intercepts

The $$x$$-intercepts are determined by letting $$f(\theta) = 0$$ and solving for $$\theta$$.

The $$y$$-intercept is calculated by letting $$\theta = 0$$ and solving for $$f(\theta)$$. \begin{align*} y &= \tan k\theta \\ &= \tan \text{0}\text{°} \\ &= 0 \end{align*} This gives the point $$(\text{0}\text{°};0)$$.

Asymptotes

These are the values of $$k\theta$$ for which $$\tan k\theta$$ is undefined.

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