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The Tangent Function

5.7 The tangent function (EMBH8)

Revision (EMBH9)

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

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.png9f7d362606019519ade93692d8161ce7.png

\(q=0\)

4d88551659d54dd8f23f375a43e9cc6b.png1c4376363790fc45fefe75c3430c9a04.png

\(q<0\)

c31962902b66d146ba0ebecb7dd52d15.png7aa1b8cd46f11243ddc1d0e318c68c6d.png

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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

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

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.png42f7ab84664069f0ebcc474896b509ae.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.

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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

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

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.png81165599bb0c8daca3d678a4cfaefabe.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 \((\text{0}\text{°};\tan p)\).

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

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

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

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

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

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

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