\(y_1 = \tan \theta - \frac{1}{2}\)
5.7 The tangent function
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5.6 The cosine function
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5.8 Summary
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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.
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Period: \(\text{180}\text{°}\)
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Domain: \(\left\{\theta : \text{0}\text{°} \le \theta \le \text{360}\text{°}, \theta \ne \text{90}\text{°}; \text{270}\text{°}\right\}\)
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Range: \(\left\{f(\theta):f(\theta)\in ℝ\right\}\)
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\(x\)-intercepts: \(\left(\text{0}\text{°};0\right)\), \(\left(\text{180}\text{°};0\right)\), \(\left(\text{360}\text{°};0\right)\)
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\(y\)-intercept: \(\left(\text{0}\text{°};0\right)\)
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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 \(f(\theta) = a \tan \theta + q\):
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The effect of \(q\) on vertical shift
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For \(q>0\), \(f(\theta)\) is shifted vertically upwards by \(q\) units.
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For \(q<0\), \(f(\theta)\) is shifted vertically downwards by \(q\) units.
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The effect of \(a\) on shape
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For \(a>1\), branches of \(f(\theta)\) are steeper.
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For \(0<a<1\), branches of \(f(\theta)\) are less steep and curve more.
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For \(a<0\), there is a reflection about the \(x\)-axis.
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For \(-1 < a < 0\), there is a reflection about the \(x\)-axis and the branches of the graph are less steep.
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For \(a < -1\), there is a reflection about the \(x\)-axis and the branches of the graph are steeper.
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\(a<0\) |
\(a>0\) |
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\(q>0\) |
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\(q=0\) |
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\(q<0\) |
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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
\(y_2 = - 3 \tan \theta\)
\(y_3 = \tan \theta + 2\)
\(y_4 = 2 \tan \theta - 1\)
Functions of the form \(y=\tan (k\theta)\) (EMBHB)
The effects of \(k\) on a tangent graph
- 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 \(\text{60}\)\(\text{°}\) \(\text{120}\)\(\text{°}\) \(\text{180}\)\(\text{°}\) \(\text{240}\)\(\text{°}\) \(\text{300}\)\(\text{°}\) \(\text{360}\)\(\text{°}\) \(\tan \theta\) -
Use the table of values to plot the graph of \(y_1 = \tan \theta\) for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\).
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On the same system of axes, plot the following graphs:
- \(y_2 = \tan (-\theta)\)
- \(y_3 = \tan 3\theta\)
- \(y_4 = \tan \frac{\theta}{2}\)
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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 \(k\) -
What do you notice about \(y_1 = \tan \theta\) and \(y_2 = \tan (-\theta)\)?
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Is \(\tan (-\theta) = -\tan \theta\) a true statement? Explain your answer.
- 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.
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For \(k > 0\):
For \(k > 1\), the period of the tangent function decreases.
For \(0 < k < 1\), the period of the tangent function increases.
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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 \(k\).
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\(k > 0\) |
\(k < 0\) |
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Worked example 26: Tangent function
- Sketch the following functions on the same set of axes for
\(-\text{180}\text{°} \leq \theta \leq
\text{180}\text{°}\).
- \(y_1 = \tan \theta\)
- \(y_2 = \tan \frac{3\theta}{2}\)
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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
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.
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
\(f(\theta) =\tan 2\theta\)
\(g(\theta) =\tan \frac{3\theta}{4}\)
\(h(\theta) =\tan (-2\theta)\)
\(k(\theta) =\tan \frac{2\theta}{3}\)
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
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On the same system of axes, plot the following graphs for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\):
- \(y_1 = \tan \theta\)
- \(y_2 = \tan (\theta - \text{60}\text{°})\)
- \(y_3 = \tan (\theta - \text{90}\text{°})\)
- \(y_4 = \tan (\theta + \text{60}\text{°})\)
- \(y_5 = \tan (\theta + \text{180}\text{°})\)
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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.
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For \(p > 0\), the graph of the tangent function shifts to the left by \(p\).
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For \(p < 0\), the graph of the tangent function shifts to the right by \(p\).
| \(p > 0\) | \(p < 0\) |
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Worked example 27: Tangent function
- Sketch the following functions on the same set of axes for
\(-\text{180}\text{°} \leq \theta \leq
\text{180}\text{°}\).
- \(y_1 = \tan \theta\)
- \(y_2 = \tan (\theta + \text{30}\text{°})\)
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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
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 \(y = \tan (\theta + p)\)
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
\(f(\theta) =\tan (\theta + \text{45}\text{°})\)
\(g(\theta) =\tan (\theta - \text{30}\text{°})\)
\(h(\theta) =\tan (\theta + \text{60}\text{°})\)
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
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 \(-\text{90}\text{°} \leq \theta \leq \text{90}\text{°}\)
\(f(\theta) = -\tan 2\theta\) for \(\text{0}\text{°} \leq \theta \leq \text{90}\text{°}\)
\(y = \frac{1}{2} \tan (\theta + \text{45}\text{°})\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\)
\(y = \tan (30° - \theta)\) for \(-180° \leq \theta \leq 180° \)
Given the graph of \(y = a \tan k\theta\), determine the values of \(a\) and \(k\).
\(a = -1\); \(k = \frac{1}{2}\)
Mixed exercises
Textbook Exercise 5.32
Determine the equation for each of the following:
\(f(\theta) = a \sin k\theta\) and \(g(\theta) = a \tan \theta\)
\(f(\theta) = \frac{3}{2} \sin 2\theta\) 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 \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\).
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5.6 The cosine function
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5.8 Summary
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