\(y_1 = \tan \theta - \frac{1}{2}\)
5.7 The tangent function
Previous
5.6 The cosine function
|
Next
5.8 Summary
|
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.
-
\(a<0\) |
\(a>0\) |
|
\(q>0\) |
||
\(q=0\) |
||
\(q<0\) |
Revision
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 \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\).
-
On the same system of axes, plot the following graphs:
- \(y_2 = \tan (-\theta)\)
- \(y_3 = \tan 3\theta\)
- \(y_4 = \tan \frac{\theta}{2}\)
-
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 \(y_2 = \tan (-\theta)\)?
-
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.
-
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
- Sketch the following functions on the same set of axes for
\(-\text{180}\text{°} \leq \theta \leq
\text{180}\text{°}\).
- \(y_1 = \tan \theta\)
- \(y_2 = \tan \frac{3\theta}{2}\)
-
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 \(y = \tan k\theta\)
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
-
On the same system of axes, plot the following graphs for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\):
- \(y_1 = \tan \theta\)
- \(y_2 = \tan (\theta - \text{60}\text{°})\)
- \(y_3 = \tan (\theta - \text{90}\text{°})\)
- \(y_4 = \tan (\theta + \text{60}\text{°})\)
- \(y_5 = \tan (\theta + \text{180}\text{°})\)
-
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
- Sketch the following functions on the same set of axes for
\(-\text{180}\text{°} \leq \theta \leq
\text{180}\text{°}\).
- \(y_1 = \tan \theta\)
- \(y_2 = \tan (\theta + \text{30}\text{°})\)
-
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 \(y = \tan (\theta + p)\)
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
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 \(k\).
Mixed exercises
Determine the equation for each of the following:
\(f(\theta) = a \sin k\theta\) and \(g(\theta) = a \tan \theta\)
\(f(\theta) = a \sin k\theta\) and \(g(\theta) = a \cos ( \theta + p)\)
\(y = a \tan k\theta\)
\(y = a \cos \theta + q\)
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 \(f\)?
What is the amplitude of \(g\)?
Use your sketch to determine how many solutions there are for the equation \(2 \sin \theta - \cos \theta = 1\). Give one of the solutions.
Indicate on your sketch where on the graph the solution to \(2 \sin \theta = -1\) is found.
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 \(c\).
What is the period of \(g\)?
Solve the equation \(\cos \theta = \frac{1}{2}\) graphically and show your answer(s) on the diagram.
Determine the equation of the new graph if \(g\) is reflected about the \(x\)-axis and shifted to the right by \(\text{45}\text{°}\).
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 \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\).
Previous
5.6 The cosine function
|
Table of Contents |
Next
5.8 Summary
|