Home Practice
For learners and parents For teachers and schools
Textbooks
Full catalogue
Leaderboards
Learners Leaderboard Classes/Grades Leaderboard Schools Leaderboard
Pricing Support
Help centre Contact us
Log in

We think you are located in United States. Is this correct?

5.7 The tangent function

5.7 The tangent function (EMBH8)

Revision (EMBH9)

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

b0314c458ecaf2909d5e63fb0e619ed2.png

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.

\(a<0\)

\(a>0\)

\(q>0\)

73b7bdb50e5399566ae55f80007c1111.png 9f7d362606019519ade93692d8161ce7.png

\(q=0\)

4d88551659d54dd8f23f375a43e9cc6b.png 1c4376363790fc45fefe75c3430c9a04.png

\(q<0\)

c31962902b66d146ba0ebecb7dd52d15.png 7aa1b8cd46f11243ddc1d0e318c68c6d.png
temp text

Revision

Textbook 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

\(y_1 = \tan \theta - \frac{1}{2}\)

c2551fe8184026feaa6e04cc8461d542.png

\(y_2 = - 3 \tan \theta\)

2d65bfb309d657d1a3e53483fc8b12e7.png

\(y_3 = \tan \theta + 2\)

68fbe7578e28e313ef7e0ed1fe3233ae.png

\(y_4 = 2 \tan \theta - 1\)

3097a05401c67d14cc6a7609839b63f1.png

Functions of the form \(y=\tan (k\theta)\) (EMBHB)

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 \(k\).

\(k > 0\)

\(k < 0\)

a985421ea6ffa4aa20941a6608d9107e.png 42f7ab84664069f0ebcc474896b509ae.png

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

Sketch the tangent graphs

c372c20878f94421fff696d0d35df0ba.png

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.

temp text

Tangent functions of the form \(y = \tan k\theta\)

Textbook 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

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

888b352e03186e99d031dad58a48ff71.png

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

30745ce501dc2815299a70cca8a28ded.png

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

817cc87f7bf7bc41d7355df961d223a0.png

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

6b435e59ba5fe41dd5139ec5f19cf463.png

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 \(p\).

\(p > 0\) \(p < 0\)
24f2551d9186fef1eb718838c98a567f.png 81165599bb0c8daca3d678a4cfaefabe.png

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

Sketch the tangent graphs

e56ac993e7953cc8f703f4834684efba.png

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 \((\text{0}\text{°};\tan p)\).

temp text

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

Textbook 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

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

d5b449c834852cbf914f759e5ad50476.png

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

feec2a59547a869ae6c61da071ca5055.png

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

3af5e52bd9e43862d663070ba33768e8.png

Sketching tangent graphs (EMBHD)

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

3ac1ebf912ed63f232f92f1dd79d6f0d.png

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

Textbook Exercise 5.31

Sketch the following graphs on separate axes:

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

ffa518f48dba6e17f5c69e775aa1f21f.png

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

a82f2460e39eecf1f7d4716a87b42134.png

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

668df705b606fb69128a3ac63b84e344.png

\(y = \tan (30° - \theta)\) for \(-180° \leq \theta \leq 180° \)

50df4216a08cb6615e0b66dab426c2f9.png

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

6676cff0eda4d94dbeab67c5afc897d6.png
\(a = -1\); \(k = \frac{1}{2}\)

Mixed exercises

Textbook Exercise 5.32

Determine the equation for each of the following:

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

c92a9dc32d36fe051e29c108ead2d2f8.png
\(f(\theta) = \frac{3}{2} \sin 2\theta\) and \(g(\theta) = -\frac{3}{2} \tan \theta\)

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

93761936c924ee9e3ba3de0cdcc35016.png
\(f(\theta) = -2 \sin \theta\) and \(g(\theta) = 2 \cos (\theta + \text{360}\text{°})\)

\(y = a \tan k\theta\)

87e5dca873444fcea6ffef19fada2434.png
\(y = 3 \tan \frac{\theta}{2}\)

\(y = a \cos \theta + q\)

c29084d2592f2b354690060920643be2.png
\(y = y = 2 \cos \theta + 2\)

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.

74b0509069a5a2bc413e3adde30c4ff5.png

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.

e4ef1d0177c0cdbb53051708cfc66fc8.png

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 \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\).

11facd49d2257ba3b2105e5830a0cecd.png