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# The tangent function

## 5.7 The tangent function (EMBH8)

### Revision (EMBH9)

#### Functions of the form $$y = \tan\theta$$ for $$\text{0}\text{°} \leq \theta \leq \text{360}\text{°}$$

The dashed vertical lines are called the asymptotes. The asymptotes are at the values of θ where $$\tan\theta$$ is not defined.

• Period: $$\text{180}\text{°}$$

• Domain: $$\left\{\theta : \text{0}\text{°} \le \theta \le \text{360}\text{°}, \theta \ne \text{90}\text{°}; \text{270}\text{°}\right\}$$

• Range: $$\left\{f(\theta):f(\theta)\in ℝ\right\}$$

• $$x$$-intercepts: $$\left(\text{0}\text{°};0\right)$$, $$\left(\text{180}\text{°};0\right)$$, $$\left(\text{360}\text{°};0\right)$$

• $$y$$-intercept: $$\left(\text{0}\text{°};0\right)$$

• Asymptotes: the lines $$\theta =\text{90}\text{°}$$ and $$\theta =\text{270}\text{°}$$

#### Functions of the form $$y = a \tan \theta + q$$

Tangent functions of the general form $$y = a \tan \theta + q$$, where $$a$$ and $$q$$ are constants.

The effects of $$a$$ and $$q$$ on $$f(\theta) = a \tan \theta + q$$:

• The effect of $$q$$ on vertical shift

• For $$q>0$$, $$f(\theta)$$ is shifted vertically upwards by $$q$$ units.

• For $$q<0$$, $$f(\theta)$$ is shifted vertically downwards by $$q$$ units.

• The effect of $$a$$ on shape

• For $$a>1$$, branches of $$f(\theta)$$ are steeper.

• For $$0<a<1$$, branches of $$f(\theta)$$ are less steep and curve more.

• For $$a<0$$, there is a reflection about the $$x$$-axis.

• For $$-1 < a < 0$$, there is a reflection about the $$x$$-axis and the branches of the graph are less steep.

• For $$a < -1$$, there is a reflection about the $$x$$-axis and the branches of the graph are steeper.

 $$a<0$$ $$a>0$$ $$q>0$$ $$q=0$$ $$q<0$$

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## Revision

Exercise 5.28

On separate axes, accurately draw each of the following functions for $$\text{0}\text{°} \leq \theta \leq \text{360}\text{°}$$:

• Use tables of values if necessary.
• Use graph paper if available.

For each function determine the following:

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

$$y_1 = \tan \theta - \frac{1}{2}$$

$$y_2 = - 3 \tan \theta$$

$$y_3 = \tan \theta + 2$$

$$y_4 = 2 \tan \theta - 1$$

## 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. $$y_4 = \tan \frac{\theta}{2}$$
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, $\text{Period} = \frac{\text{180}\text{°}}{|k|}$ where $$|k|$$ is the absolute value of $$k$$.

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

## Worked example 26: Tangent function

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. $$y_2 = \tan \frac{3\theta}{2}$$
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 $$y_2 = \tan \frac{3\theta}{2}$$, 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 $$-\text{1}$$ $$\text{0}$$ $$\tan \frac{3\theta}{2}$$ undef $$-\text{0,41}$$ $$\text{1}$$ $$-\text{2,41}$$ $$\text{0}$$ $$\text{2,41}$$ $$-\text{1}$$ $$\text{0,41}$$ undef

### Complete the table

 $$y_1 = \tan \theta$$ $$y_2 = \tan \frac{3\theta}{2}$$ 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 $$\{ \theta: -\frac{\text{90}\text{°}}{k} < \theta < \frac{\text{90}\text{°}}{k}, \theta \in \mathbb{R}\}$$ because $$f(\theta)$$ is undefined for $$\theta = -\frac{\text{90}\text{°}}{k}$$ and $$\theta = \frac{\text{90}\text{°}}{k}$$.

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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## Tangent functions of the form $$y = \tan k\theta$$

Exercise 5.29

Sketch the following functions for $$-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}$$. For each graph determine:

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

$$f(\theta) =\tan 2\theta$$

$$g(\theta) =\tan \frac{3\theta}{4}$$

$$h(\theta) =\tan (-2\theta)$$

$$k(\theta) =\tan \frac{2\theta}{3}$$

### Functions of the form $$y=\tan\left(\theta +p\right)$$ (EMBHC)

We now consider tangent functions of the form $$y = \tan(\theta + p)$$ and the effects of parameter $$p$$.

## The effects of $$p$$ on a tangent graph

1. On the same system of axes, plot the following graphs for $$-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}$$:

1. $$y_1 = \tan \theta$$
2. $$y_2 = \tan (\theta - \text{60}\text{°})$$
3. $$y_3 = \tan (\theta - \text{90}\text{°})$$
4. $$y_4 = \tan (\theta + \text{60}\text{°})$$
5. $$y_5 = \tan (\theta + \text{180}\text{°})$$
2. Use your sketches of the functions above to complete the following table:

 $$y_1$$ $$y_2$$ $$y_3$$ $$y_4$$ $$y_5$$ period domain range $$y$$-intercept(s) $$x$$-intercept(s) asymptotes effect of $$p$$

The effect of the parameter on $$y = \tan(\theta + p)$$

The effect of $$p$$ on the tangent function is a horizontal shift (or phase shift); the entire graph slides to the left or to the right.

• For $$p > 0$$, the graph of the tangent function shifts to the left by $$p$$.

• For $$p < 0$$, the graph of the tangent function shifts to the right by $$p$$.

 $$p > 0$$ $$p < 0$$

## Worked example 27: Tangent function

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. $$y_2 = \tan (\theta + \text{30}\text{°})$$
2. For each function determine the following:

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

### Examine the equations of the form $$y = \tan (\theta + p)$$

Notice that for $$y_1 = \tan \theta$$ we have $$p = \text{0}\text{°}$$ (no phase shift) and for $$y_2 = \tan (\theta + \text{30}\text{°})$$ we have $$p = \text{30}\text{°} > 0$$ and therefore the graph shifts to the left by $$\text{30}$$$$\text{°}$$.

### 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 $$-\text{1}$$ $$\text{0}$$ $$\tan (\theta + \text{30}\text{°})$$ $$\text{0,58}$$ $$\text{3,73}$$ $$-\text{1,73}$$ $$-\text{0,27}$$ $$\text{0,58}$$ $$\text{3,73}$$ $$-\text{1,73}$$ $$-\text{0,27}$$ $$\text{0,58}$$

### Complete the table

 $$y_1 = \tan \theta$$ $$y_2 = \tan (\theta + \text{30}\text{°})$$ period $$\text{180}\text{°}$$ $$\text{180}\text{°}$$ domain $$\{ \theta: -\text{180}\text{°} \leq \theta \leq \text{180}\text{°}, \theta \ne -\text{90}\text{°}; \text{90}\text{°} \}$$ $$\{ \theta: -\text{180}\text{°} \leq \theta \leq \text{180}\text{°}, \theta \ne -\text{120}\text{°}; \text{60}\text{°} \}$$ range $$(-\infty;\infty)$$ $$(-\infty;\infty)$$ $$y$$-intercept(s) $$(\text{0}\text{°};0)$$ $$(\text{0}\text{°};\text{0,58})$$ $$x$$-intercept(s) $$(-\text{180}\text{°};0)$$, $$(\text{0}\text{°};0)$$ and $$(\text{180}\text{°};0)$$ $$(-\text{30}\text{°};0) \text{ and } (\text{150}\text{°};0)$$ asymptotes $$\theta = -\text{90}\text{°} \text{ and } \theta = \text{90}\text{°}$$ $$\theta = -\text{120}\text{°} \text{ and } \theta = \text{60}\text{°}$$

#### Discovering the characteristics

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

Domain and range

The domain of one branch is $$\{ \theta: \theta \in (-\text{90}\text{°} - p; \text{90}\text{°} - p) \}$$ because the function is undefined for $$\theta = -\text{90}\text{°} - p$$ and $$\theta = \text{90}\text{°} - p$$.

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

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 &= \tan (\theta + p) \\ &= \tan (\text{0}\text{°} + p) \\ &= \tan p \end{align*} This gives the point $$(\text{0}\text{°};\tan p)$$.

## Tangent functions of the form $$y = \tan (\theta + p)$$

Exercise 5.30

Sketch the following functions for $$-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}$$.

For each function, determine the following:

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

$$f(\theta) =\tan (\theta + \text{45}\text{°})$$

$$g(\theta) =\tan (\theta - \text{30}\text{°})$$

$$h(\theta) =\tan (\theta + \text{60}\text{°})$$

## Worked example 28: Sketching a tangent graph

Sketch the graph of $$f(\theta) = \tan \frac{1}{2}(\theta - \text{30}\text{°})$$ for $$-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}$$.

### Examine the form of the equation

From the equation we see that $$0 < k < 1$$, therefore the branches of the graph will be less steep than the standard tangent graph $$y = \tan \theta$$. We also notice that $$p < 0$$ so the graph will be shifted to the right on the $$x$$-axis.

### Determine the period

The period for $$f(\theta) = \tan \frac{1}{2}(\theta - \text{30}\text{°})$$ is:

\begin{align*} \text{Period} &= \frac{\text{180}\text{°}}{|k|} \\ &= \dfrac{\text{180}\text{°}}{\frac{1}{2}} \\ &= \text{360}\text{°} \end{align*}

### Determine the asymptotes

The standard tangent graph, $$y = \tan \theta$$, for $$-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}$$ is undefined at $$\theta = -\text{90}\text{°}$$ and $$\theta = \text{90}\text{°}$$. Therefore we can determine the asymptotes of $$f(\theta) = \tan \frac{1}{2}(\theta - \text{30}\text{°})$$:

• $$\frac{-\text{90}\text{°}}{\text{0,5}} + \text{30}\text{°} = -\text{150}\text{°}$$
• $$\frac{\text{90}\text{°}}{\text{0,5}} + \text{30}\text{°} = \text{210}\text{°}$$

The asymptote at $$\theta = \text{210}\text{°}$$ lies outside the required interval.

### Plot the points and join with a smooth curve

Period: $$\text{360}\text{°}$$

Domain: $$\{ \theta: -\text{180}\text{°} \leq \theta \leq \text{180}\text{°}, \theta \ne -\text{150}\text{°} \}$$

Range: $$(-\infty;\infty)$$

$$y$$-intercepts: $$(\text{0}\text{°};-\text{0,27})$$

$$x$$-intercept: $$(\text{30}\text{°};0)$$

Asymptotes: $$\theta = -\text{150}\text{°}$$

## The tangent function

Exercise 5.31

Sketch the following graphs on separate axes:

$$y = \tan \theta - 1$$ for $$-\text{90}\text{°} \leq \theta \leq \text{90}\text{°}$$

$$f(\theta) = -\tan 2\theta$$ for $$\text{0}\text{°} \leq \theta \leq \text{90}\text{°}$$

$$y = \frac{1}{2} \tan (\theta + \text{45}\text{°})$$ for $$\text{0}\text{°} \leq \theta \leq \text{360}\text{°}$$

$$y = \tan (\text{30}\text{°} - \theta)$$ for $$-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}$$

Given the graph of $$y = a \tan k\theta$$, determine the values of $$a$$ and $$k$$.

$$a = -1$$; $$k = \frac{1}{2}$$

## Mixed exercises

Exercise 5.32

Determine the equation for each of the following:

$$f(\theta) = a \sin k\theta$$ and $$g(\theta) = a \tan \theta$$

$$f(\theta) = \frac{3}{2} \sin 2\theta$$ and $$g(\theta) = -\frac{3}{2} \tan \theta$$

$$f(\theta) = a \sin k\theta$$ and $$g(\theta) = a \cos ( \theta + p)$$

$$f(\theta) = -2 \sin \theta$$ and $$g(\theta) = 2 \cos (\theta + \text{360}\text{°})$$

$$y = a \tan k\theta$$

$$y = 3 \tan \frac{\theta}{2}$$

$$y = a \cos \theta + q$$

$$y = y = 2 \cos \theta + 2$$

Given the functions $$f(\theta) = 2 \sin \theta$$ and $$g(\theta) = \cos \theta + 1$$:

Sketch the graphs of both functions on the same system of axes, for $$\text{0}\text{°} \leq \theta \leq \text{360}\text{°}$$. Indicate the turning points and intercepts on the diagram.

What is the period of $$f$$?

$$\text{360}$$$$\text{°}$$

What is the amplitude of $$g$$?

$$\text{1}$$

Use your sketch to determine how many solutions there are for the equation $$2 \sin \theta - \cos \theta = 1$$. Give one of the solutions.

At $$\theta = \text{180}\text{°}$$

Indicate on your sketch where on the graph the solution to $$2 \sin \theta = -1$$ is found.

todo

The sketch shows the two functions $$f(\theta) = a \cos \theta$$ and $$g(\theta) = \tan \theta$$ for $$\text{0}\text{°} \leq \theta \leq \text{360}\text{°}$$. Points $$P(\text{135}\text{°}; b)$$ and $$Q(c; -1)$$ lie on $$g(\theta)$$ and $$f(\theta)$$ respectively.

Determine the values of $$a$$, $$b$$ and $$c$$.

$$a = 2$$, $$b = -1$$ and $$c = \text{240}\text{°}$$

What is the period of $$g$$?

$$\text{180}\text{°}$$

Solve the equation $$\cos \theta = \frac{1}{2}$$ graphically and show your answer(s) on the diagram.

$$\theta = \text{60}\text{°}; \text{300}\text{°}$$

Determine the equation of the new graph if $$g$$ is reflected about the $$x$$-axis and shifted to the right by $$\text{45}\text{°}$$.

$$y = - \tan (\theta - \text{45}\text{°})$$

Sketch the graphs of $$y_1 = -\frac{1}{2} \sin (\theta + \text{30}\text{°})$$ and $$y_2 = \cos (\theta - \text{60}\text{°})$$, on the same system of axes for $$\text{0}\text{°} \leq \theta \leq \text{360}\text{°}$$.